Conditional Statement – Definition, Truth Table, Examples, FAQs
What is a conditional statement, how to write a conditional statement, what is a biconditional statement, solved examples on conditional statements, practice problems on conditional statements, frequently asked questions about conditional statements.
A conditional statement is a statement that is written in the “If p, then q” format. Here, the statement p is called the hypothesis and q is called the conclusion. It is a fundamental concept in logic and mathematics.
Conditional statement symbol : p → q
A conditional statement consists of two parts.
- The “if” clause, which presents a condition or hypothesis.
- The “then” clause, which indicates the consequence or result that follows if the condition is true.
Example : If you brush your teeth, then you won’t get cavities.
Hypothesis (Condition): If you brush your teeth
Conclusion (Consequence): then you won’t get cavities
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Conditional Statement: Definition
A conditional statement is characterized by the presence of “if” as an antecedent and “then” as a consequent. A conditional statement, also known as an “if-then” statement consists of two parts:
- The “if” clause (hypothesis): This part presents a condition, situation, or assertion. It is the initial condition that is being considered.
- The “then” clause (conclusion): This part indicates the consequence, result, or action that will occur if the condition presented in the “if” clause is true or satisfied.
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Representation of Conditional Statement
The conditional statement of the form ‘If p, then q” is represented as p → q.
It is pronounced as “p implies q.”
Different ways to express a conditional statement are:
- p implies q
- p is sufficient for q
- q is necessary for p
Parts of a Conditional Statement
There are two parts of conditional statements, hypothesis and conclusion. The hypothesis or condition will begin with the “if” part, and the conclusion or action will begin with the “then” part. A conditional statement is also called “implication.”
Conditional Statements Examples:
Example 1: If it is Sunday, then you can go to play.
Hypothesis: If it is Sunday
Conclusion: then you can go to play.
Example 2: If you eat all vegetables, then you can have the dessert.
Condition: If you eat all vegetables
Conclusion: then you can have the dessert
To form a conditional statement, follow these concise steps:
Step 1 : Identify the condition (antecedent or “if” part) and the consequence (consequent or “then” part) of the statement.
Step 2 : Use the “if… then…” structure to connect the condition and consequence.
Step 3 : Ensure the statement expresses a logical relationship where the condition leads to the consequence.
Example 1 : “If you study (condition), then you will pass the exam (consequence).”
This conditional statement asserts that studying leads to passing the exam. If you study (condition is true), then you will pass the exam (consequence is also true).
Example 2 : If you arrange the numbers from smallest to largest, then you will have an ascending order.
Hypothesis: If you arrange the numbers from smallest to largest
Conclusion: then you will have an ascending order
Truth Table for Conditional Statement
The truth table for a conditional statement is a table used in logic to explore the relationship between the truth values of two statements. It lists all possible combinations of truth values for “p” and “q” and determines whether the conditional statement is true or false for each combination.
The truth value of p → q is false only when p is true and q is False.
If the condition is false, the consequence doesn’t affect the truth of the conditional; it’s always true.
In all the other cases, it is true.
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | T |
The truth table is helpful in the analysis of possible combinations of truth values for hypothesis or condition and conclusion or action. It is useful to understand the presence of truth or false statements.
Converse, Inverse, and Contrapositive
The converse, inverse, and contrapositive are three related conditional statements that are derived from an original conditional statement “p → q.”
| Conditional Statement | p q |
| Converse | q p |
| Inverse | ~p → ~q |
| Contrapositive | ~q → ~p |
Consider a conditional statement: If I run, then I feel great.
- Converse:
The converse of “p → q” is “q → p.” It reverses the order of the original statement. While the original statement says “if p, then q,” the converse says “if q, then p.”
Converse: If I feel great, then I run.
- Inverse:
The inverse of “p → q” is “~p → ~q,” where “” denotes negation (opposite). It negates both the antecedent (p) and the consequent (q). So, if the original statement says “if p, then q,” the inverse says “if not p, then not q.”
Inverse : If I don’t run, then I don’t feel great.
- Contrapositive:
The contrapositive of “p → q” is “~q → ~p.” It reverses the order and also negates both the statements. So, if the original statement says “if p, then q,” the contrapositive says “if not q, then not p.”
Contrapositive: If I don’t feel great, then I don’t run.
A biconditional statement is a type of compound statement in logic that expresses a bidirectional or two-way relationship between two statements. It asserts that “p” is true if and only if “q” is true, and vice versa. In symbolic notation, a biconditional statement is represented as “p ⟺ q.”
In simpler terms, a biconditional statement means that the truth of “p” and “q” are interdependent.
If “p” is true, then “q” must also be true, and if “q” is true, then “p” must be true. Conversely, if “p” is false, then “q” must be false, and if “q” is false, then “p” must be false.
Biconditional statements are often used to express equality, equivalence, or conditions where two statements are mutually dependent for their truth values.
Examples :
- I will stop my bike if and only if the traffic light is red.
- I will stay if and only if you play my favorite song.
Facts about Conditional Statements
- The negation of a conditional statement “p → q” is expressed as “p and not q.” It is denoted as “𝑝 ∧ ∼𝑞.”
- The conditional statement is not logically equivalent to its converse and inverse.
- The conditional statement is logically equivalent to its contrapositive.
- Thus, we can write p → q ∼q → ∼p
In this article, we learned about the fundamentals of conditional statements in mathematical logic, including their structure, parts, truth tables, conditional logic examples, and various related concepts. Understanding conditional statements is key to logical reasoning and problem-solving. Now, let’s solve a few examples and practice MCQs for better comprehension.
Example 1: Identify the hypothesis and conclusion.
If you sing, then I will dance.
Solution :
Given statement: If you sing, then I will dance.
Here, the antecedent or the hypothesis is “if you sing.”
The conclusion is “then I will dance.”
Example 2: State the converse of the statement: “If the switch is off, then the machine won’t work.”
Here, p: The switch is off
q: The machine won’t work.
The conditional statement can be denoted as p → q.
Converse of p → q is written by reversing the order of p and q in the original statement.
Converse of p → q is q → p.
Converse of p → q: q → p: If the machine won’t work, then the switch is off.
Example 3: What is the truth value of the given conditional statement?
If 2+2=5 , then pigs can fly.
Solution:
q: Pigs can fly.
The statement p is false. Now regardless of the truth value of statement q, the overall statement will be true.
F → F = T
Hence, the truth value of the statement is true.
Conditional Statement - Definition, Truth Table, Examples, FAQs
Attend this quiz & Test your knowledge.
What is the antecedent in the given conditional statement? If it’s sunny, then I’ll go to the beach.
A conditional statement can be expressed as, what is the converse of “a → b”, when the antecedent is true and the consequent is false, the conditional statement is.
What is the meaning of conditional statements?
Conditional statements, also known as “if-then” statements, express a cause-and-effect or logical relationship between two propositions.
When does the truth value of a conditional statement is F?
A conditional statement is considered false when the antecedent is true and the consequent is false.
What is the contrapositive of a conditional statement?
The contrapositive reverses the order of the statements and also negates both the statements. It is equivalent in truth value to the original statement.
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Conditional Statement If Then's Defined in Geometry - 15+ Examples!
// Last Updated: January 21, 2020 - Watch Video //
In today’s geometry lesson , you’re going to learn all about conditional statements!
Jenn, Founder Calcworkshop ® , 15+ Years Experience (Licensed & Certified Teacher)
We’re going to walk through several examples to ensure you know what you’re doing.
In addition, this lesson will prepare you for deductive reasoning and two column proofs later on.
Here we go!
What are Conditional Statements?
To better understand deductive reasoning, we must first learn about conditional statements.
A conditional statement has two parts: hypothesis ( if ) and conclusion ( then ).
In fact, conditional statements are nothing more than “If-Then” statements!
Sometimes a picture helps form our hypothesis or conclusion. Therefore, we sometimes use Venn Diagrams to visually represent our findings and aid us in creating conditional statements.
But to verify statements are correct, we take a deeper look at our if-then statements. This is why we form the converse , inverse , and contrapositive of our conditional statements.
What is the Converse of a Statement?
Well, the converse is when we switch or interchange our hypothesis and conclusion.
Conditional Statement : “If today is Wednesday, then yesterday was Tuesday.”
Hypothesis : “If today is Wednesday” so our conclusion must follow “Then yesterday was Tuesday.”
So the converse is found by rearranging the hypothesis and conclusion, as Math Planet accurately states.
Converse : “If yesterday was Tuesday, then today is Wednesday.”
What is the Inverse of a Statement?
Now the inverse of an If-Then statement is found by negating (making negative) both the hypothesis and conclusion of the conditional statement.
So using our current conditional statement, “If today is Wednesday, then yesterday was Tuesday”.
Inverse : “If today is not Wednesday, then yesterday was not Tuesday.”
What is a Contrapositive?
And the contrapositive is formed by interchanging the hypothesis and conclusion and then negating both.
Contrapositive : “If yesterday was not Tuesday, then today is not Wednesday”
What is a Biconditional Statement?
A statement written in “if and only if” form combines a reversible statement and its true converse. In other words the conditional statement and converse are both true.
Continuing with our initial condition, “If today is Wednesday, then yesterday was Tuesday.”
Biconditional : “Today is Wednesday if and only if yesterday was Tuesday.”
Examples of Conditional Statements
In the video below we will look at several harder examples of how to form a proper statement, converse, inverse, and contrapositive. And here’s a big hint…
Whenever you see “con” that means you switch! It’s like being a con-artist!
Moreover, we will detail the process for coming up with reasons for our conclusions using known postulates. We will review the ten postulates that we have learned so far, and add a few more problems dealing with perpendicular lines, planes, and perpendicular bisectors.
After this lesson, we will be ready to tackle deductive reasoning head-on, and feel confident as we march onward toward learning two-column proofs!
Conditional Statements – Lesson & Examples (Video)
- Introduction to conditional statements
- 00:00:25 – What are conditional statements, converses, and biconditional statements? (Examples #1-2)
- 00:05:21 – Understanding venn diagrams (Examples #3-4)
- 00:11:07 – Supply the missing venn diagram and conditional statement for each question (Examples #5-8)
- Exclusive Content for Member’s Only
- 00:17:48 – Write the statement and converse then determine if they are reversible (Examples #9-12)
- 00:29:17 – Understanding the inverse, contrapositive, and symbol notation
- 00:35:33 – Write the statement, converse, inverse, contrapositive, and biconditional statements for each question (Examples #13-14)
- 00:45:40 – Using geometry postulates to verify statements (Example #15)
- 00:53:23 – What are perpendicular lines, perpendicular planes and the perpendicular bisector?
- 00:56:26 – Using the figure, determine if the statement is true or false (Example #16)
- Practice Problems with Step-by-Step Solutions
- Chapter Tests with Video Solutions
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Conditional Statement
A conditional statement is a part of mathematical reasoning which is a critical skill that enables students to analyze a given hypothesis without any reference to a particular context or meaning. In layman words, when a scientific inquiry or statement is examined, the reasoning is not based on an individual's opinion. Derivations and proofs need a factual and scientific basis.
Mathematical critical thinking and logical reasoning are important skills that are required to solve maths reasoning questions.
In this mini-lesson, we will explore the world of conditional statements. We will walk through the answers to the questions like what is meant by a conditional statement, what are the parts of a conditional statement, and how to create conditional statements along with solved examples and interactive questions.
Lesson Plan
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What Is Meant By a Conditional Statement?
A statement that is of the form "If p, then q" is a conditional statement. Here 'p' refers to 'hypothesis' and 'q' refers to 'conclusion'.
For example, "If Cliff is thirsty, then she drinks water."
This is a conditional statement. It is also called an implication.
'\(\rightarrow\)' is the symbol used to represent the relation between two statements. For example, A\(\rightarrow\)B. It is known as the logical connector. It can be read as A implies B.
Here are two more conditional statement examples
Example 1: If a number is divisible by 4, then it is divisible by 2.
Example 2: If today is Monday, then yesterday was Sunday.
What Are the Parts of a Conditional Statement?
Hypothesis (if) and Conclusion (then) are the two main parts that form a conditional statement.
Let us consider the above-stated example to understand the parts of a conditional statement.
Conditional Statement : If today is Monday, then yesterday was Sunday.
Hypothesis : "If today is Monday."
Conclusion : "Then yesterday was Sunday."
On interchanging the form of statement the relationship gets changed.
To check whether the statement is true or false here, we have subsequent parts of a conditional statement. They are:
- Contrapositive
Biconditional Statement
Let us consider hypothesis as statement A and Conclusion as statement B.
Following are the observations made:
Converse of Statement
When hypothesis and conclusion are switched or interchanged, it is termed as converse statement . For example,
Conditional Statement : “If today is Monday, then yesterday was Sunday.”
Hypothesis : “If today is Monday”
Converse : “If yesterday was Sunday, then today is Monday.”
Here the conditional statement logic is, If B, then A (B → A)
Inverse of Statement
When both the hypothesis and conclusion of the conditional statement are negative, it is termed as an inverse of the statement. For example,
Conditional Statement: “If today is Monday, then yesterday was Sunday”.
Inverse : “If today is not Monday, then yesterday was not Sunday.”
Here the conditional statement logic is, If not A, then not B (~A → ~B)
Contrapositive Statement
When the hypothesis and conclusion are negative and simultaneously interchanged, then the statement is contrapositive. For example,
Contrapositive: “If yesterday was not Sunday, then today is not Monday”
Here the conditional statement logic is, if not B, then not A (~B → ~A)
The statement is a biconditional statement when a statement satisfies both the conditions as true, being conditional and converse at the same time. For example,
Biconditional : “Today is Monday if and only if yesterday was Sunday.”
Here the conditional statement logic is, A if and only if B (A ↔ B)
How to Create Conditional Statements?
Here, the point to be kept in mind is that the 'If' and 'then' part must be true.
If a number is a perfect square , then it is even.
- 'If' part is a number that is a perfect square.
Think of 4 which is a perfect square.
This has become true.
- The 'then' part is that the number should be even. 4 is even.
This has also become true.
Thus, we have set up a conditional statement.
Let us hypothetically consider two statements, statement A and statement B. Observe the truth table for the statements:
|
| ||
| Truth | Truth | Truth |
| Truth | False | False |
| False | Truth | Truth |
| False | False | Truth |
According to the table, only if the hypothesis (A) is true and the conclusion (B) is false then, A → B will be false, or else A → B will be true for all other conditions.
- A sentence needs to be either true or false, but not both, to be considered as a mathematically accepted statement.
- Any sentence which is either imperative or interrogative or exclamatory cannot be considered a mathematically validated statement.
- A sentence containing one or many variables is termed as an open statement. An open statement can become a statement if the variables present in the sentence are replaced by definite values.
Solved Examples
Let us have a look at a few solved examples on conditional statements.
Identify the types of conditional statements.
There are four types of conditional statements:
- If condition
- If-else condition
- Nested if-else
- If-else ladder.
Ray tells "If the perimeter of a rectangle is 14, then its area is 10."
Which of the following could be the counterexamples? Justify your decision.
a) A rectangle with sides measuring 2 and 5
b) A rectangle with sides measuring 10 and 1
c) A rectangle with sides measuring 1 and 5
d) A rectangle with sides measuring 4 and 3
a) Rectangle with sides 2 and 5: Perimeter = 14 and area = 10
Both 'if' and 'then' are true.
b) Rectangle with sides 10 and 1: Perimeter = 22 and area = 10
'If' is false and 'then' is true.
c) Rectangle with sides 1 and 5: Perimeter = 12 and area = 5
Both 'if' and 'then' are false.
d) Rectangle with sides 4 and 3: Perimeter = 14 and area = 12
'If' is true and 'then' is false.
Joe examined the set of numbers {16, 27, 24} to check if they are the multiples of 3. He claimed that they are divisible by 9. Do you agree or disagree? Justify your answer.
Conditional statement : If a number is a multiple of 3, then it is divisible by 9.
Let us find whether the conditions are true or false.
a) 16 is not a multiple of 3. Thus, the condition is false.
16 is not divisible by 9. Thus, the conclusion is false.
b) 27 is a multiple of 3. Thus, the condition is true.
27 is divisible by 9. Thus, the conclusion is true.
c) 24 is a multiple of 3. Thus the condition is true.
24 is not divisible by 9. Thus the conclusion is false.
Write the converse, inverse, and contrapositive statement for the following conditional statement.
If you study well, then you will pass the exam.
The given statement is - If you study well, then you will pass the exam.
It is of the form, "If p, then q"
The converse statement is, "You will pass the exam if you study well" (if q, then p).
The inverse statement is, "If you do not study well then you will not pass the exam" (if not p, then not q).
The contrapositive statement is, "If you did not pass the exam, then you did not study well" (if not q, then not p).
Interactive Questions
Here are a few activities for you to practice. Select/Type your answer and click the "Check Answer" button to see the result.
Let's Summarize
The mini-lesson targeted the fascinating concept of the conditional statement. The math journey around conditional statements started with what a student already knew and went on to creatively crafting a fresh concept in the young minds. Done in a way that not only it is relatable and easy to grasp, but also will stay with them forever.
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Be it worksheets, online classes, doubt sessions, or any other form of relation, it’s the logical thinking and smart learning approach that we, at Cuemath, believe in.
FAQs on Conditional Statement
1. what is the most common conditional statement.
'If and then' is the most commonly used conditional statement.
2. When do you use a conditional statement?
Conditional statements are used to justify the given condition or two statements as true or false.
3. What is if and if-else statement?
If is used when a specified condition is true. If-else is used when a particular specified condition is not satisfying and is false.
4. What is the symbol for a conditional statement?
'\(\rightarrow\)' is the symbol used to represent the relation between two statements. For example, A\(\rightarrow\)B. It is known as the logical connector. It can be read as A implies B.
5. What is the Contrapositive of a conditional statement?
If not B, then not A (~B → ~A)
6. What is a universal conditional statement?
Conditional statements are those statements where a hypothesis is followed by a conclusion. It is also known as an " If-then" statement. If the hypothesis is true and the conclusion is false, then the conditional statement is false. Likewise, if the hypothesis is false the whole statement is false. Conditional statements are also termed as implications.
Conditional Statement: If today is Monday, then yesterday was Sunday
Hypothesis: "If today is Monday."
Conclusion: "Then yesterday was Sunday."
If A, then B (A → B)
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Introduction to Proofs: An Active Exploration of Mathematical Language
Jennifer Firkins Nordstrom
Search Results:
Section 2.3 conditional statements, activity 2.3.1 . which type., activity 2.3.2 . relationship between universal and conditional., example 2.3.1 . universal conditional statement..
Translate the statement using quantifiers and variables, “If an integer is even then it is divisible by 2.” Answer . Let \(P(x)\) be “ \(x\) is even” and \(Q(x)\) be “ \(x\) is divisible by 2.” \(\forall x\in \mathbb{Z}, P(x)\rightarrow Q(x)\text{.}\)
Activity 2.3.3 . Translating to Universal Conditional.
Activity 2.3.4 . a geography conditional., activity 2.3.5 . a weather conditional., activity 2.3.6 . an argument conditional., activity 2.3.7 . a mathematical conditional., logical equivalences for conditionals..
- \(\displaystyle p\rightarrow q\equiv \neg p\ \vee q\)
- \(\displaystyle \neg(p\rightarrow q)\equiv p\ \wedge \neg q\)
Definition 2.3.2 .
- contrapositive : \(\neg q\rightarrow \neg p\text{;}\)
- converse : \(q\rightarrow p\text{;}\)
- inverse : \(\neg p\rightarrow \neg q\text{.}\)
Definition 2.3.3 .
Definition 2.3.4 ., activity 2.3.8 . writing contrapositives., activity 2.3.9 . more writing contrapositives., activity 2.3.10 . converse statements., converting an argument to a conditional statement..
| A |
| B |
| \(\therefore\ \)C |
Example 2.3.5 . Converting and Argument to a Conditional.
| \(p\wedge q\) |
| \(\therefore\ \)\(p\) |
Example 2.3.6 . Converting and Argument to a Conditional.
| \(p\vee q\) |
| \(\therefore\ \)\(p\) |
Activity 2.3.11 . Checking Validity with Conditional.
| \(p\rightarrow q\) |
| \(\neg q\) |
| \(\therefore\ \)\(\neg p\) |
Activity 2.3.12 . More Checking Validity with Conditional.
| \(p\rightarrow q\) |
| \(\neg p\) |
| \(\therefore\ \)\(\neg q\) |
Exercises Exercises
- Given any positive real number \(r\text{,}\) the reciprocal of ___.
- For any real number \(r\text{,}\) if \(r\) is ___ then ___.
- If a real number \(r\) ___, then ___.
- Given any negative real number \(s\text{,}\) the cube root of ___.
- For any real number \(s\text{,}\) if \(s\) is ___, then ___.
- If a real number \(s\) ___, then ___.
- There are real numbers \(u\) and \(v\) with the property that \(u+v < v-u\text{.}\)
- There is a real number \(x\) such that \(x^2 < x\text{.}\)
- For all positive integers \(n\text{,}\) \(n^2\geq n\text{.}\)
- For all real numbers \(a\) and \(b\text{,}\) \(| a+b | \leq | a | + | b |\text{.}\)
- All nonzero real numbers ___.
- For all nonzero real numbers \(r\text{,}\) there is ___ for \(r\text{.}\)
- For all nonzero real numbers \(r\text{,}\) there is a real number \(s\) such that ___.
- \(\forall\) real numbers \(x\text{,}\) if \(x>3\) then \(x^2>9\text{.}\)
- \(\forall n\in \mathbb{Z}\text{,}\) if \(n\) is prime then \(n\) is odd or \(n=2\text{.}\)
- \(\forall\) integers \(n\text{,}\) if \(n\) is divisible by 6, then \(n\) is divisible by 2 and \(n\) is divisible by 3.
| \(p\) |
| \(p\rightarrow q\) |
| \(\neg q\ \vee r\) |
| \(\therefore r\) |
| \((p\ \wedge q)\rightarrow \neg r\) |
| \(p\ \vee \neg q\) |
| \(\neg q\rightarrow p\) |
| \(\therefore \neg r\) |
| \(p \ \vee q\) |
| \(\neg p\) |
| \(\therefore q\) |
| \(\neg p\rightarrow q\) |
| \(p\) |
| \(\therefore \neg q\) |
Understanding a Conditional Statement
Definition: A Conditional Statement is…
| p | q | p |
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | T |
Example 1:
| Given: | p: I do my homework. |
| q: I get my allowance. | |
| Problem: | What does p |
Example 2:
| Given: | a: The sun is made of gas. |
| b: 3 is a prime number. | |
| Problem: | Write a |
| a | b | a |
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | T |
Example 3:
| Given: | x: Gisele has a math assignment. |
| y: David owns a car. | |
| Problem: | Write x |
In the following examples, we are given the truth values of the hypothesis and the conclusion and asked to determine the truth value of the conditional.
Example 4:
| Given: | r: 8 is an odd number. | false |
| s: 9 is composite. | true | |
| Problem: | What is the truth value of r | |
Example 5:
| Given: | r: 8 is an odd number. | false |
| s: 9 is composite. | true | |
| Problem: | What is the truth value of s | |
Example 6:
| Given: | p: 7 = 49. | true |
| q: A rectangle does not have 4 sides. | false | |
| r: Harrison Ford is an American actor. | true | |
| s: A square is not a quadrilateral. | false | |
| Problem: | Write each conditional below as a sentence. Then indicate its truth value. | |
| 1. | p | is equal to 49, then a rectangle does not have 4 sides. | |
| 2. | q | ||
| 3. | p | is equal to 49, then Harrison Ford is an American actor. | |
| 4. | q | ||
| 5. | r | is not equal to 49. | |
| 6. | ~r | is equal to 49. |
Note that in item 5, the conclusion is the negation of p. Also, in item 6, the hypothesis is the negation of r.
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 What Is A Conditional Statement?In mathematics, we define statement as a declarative statement which may either be true or may be false. Often sentences that are mathematical in nature may not be a statement because we might not know what the variable represents. For example, 2x + 2 = 5. Now here we do not know what x represents thus if we substitute the value of x (let us consider that x = 3) i.e., 2 × 3 = 6. Therefore, it is a false statement. So, what is a conditional statement? In simple words, when through a statement we put a condition on something in return of something, we call it a conditional statement. For example, Mohan tells his friend that “if you do my homework, then I will pay you 50 dollars”. So what is happening here? Mohan is paying his friend 50 dollars but places a condition that if only he’s work will be completed by his friend. A conditional statement is made up of two parts. First, there is a hypothesis that is placed after “if” and before the comma and second is a conclusion that is placed after “then”. Here, the hypothesis will be “you do my homework” and the conclusion will be “I will pay you 50 dollars”. Now, this statement can either be true or may be false. We don’t know. A hypothesis is a part that is used after the 'if' and before the comma. This composes the first part of a conditional statement. For example, the statement, 'I help you get an A+ in math,' is a hypothesis because this phrase is coming in between the 'if' and the comma. So, now I hope you can spot the hypothesis in other examples of a conditional statement. Of course, you can. Here is a statement: 'If Miley gets a car, then Allie's dog will be trained,' the hypothesis here is, 'Miley gets a car.' For the statement, 'If Tom eats chocolate ice cream, then Luke eats double chocolate ice cream,' the hypothesis here is, 'Tom eats chocolate ice cream. Now it is time for you to try and locate the hypothesis for the statement, 'If the square is a rectangle, then the rectangle is a quadrilateral'? A conclusion is a part that is used after “then”. This composes the second part of a conditional statement. For example, for the statement, “I help you get an A+ in math”, the conclusion will be “you will give me 50 dollars”. The next statement was “If Miley gets a car, then Allie's dog will be trained”, the conclusion here is Allie's dog will be trained. It is the same with the next statement and for every other conditional statement. How Do We Know If A Statement Is True or False?In mathematics, the best way we can know if a statement is true or false is by writing a mathematical proof. Before writing a proof, the mathematician must find if the statement is true or false that can be done with the help of exploration and then by finding the counterexample. Once the proof is discovered, the mathematician must communicate this discovery to those who speak the language of maths. Converse, Inverse, contrapositive, And Bi-conditional StatementWe usually use the term “converse” as a verb for talking and chatting and as a noun we use it to represent a brand of footwear. But in mathematics, we use it differently. Converse and inverse are the two terms that are a connected concept in the making of a conditional statement. If we want to create the converse of a conditional statement, we just have to switch the hypothesis and the conclusion. To create the inverse of a conditional statement, we have to turn both the hypothesis and the conclusion to the negative. A contrapositive statement can be made if we first interchange the hypothesis and conclusion then make them both negative. In a bi-conditional statement, we use “if and only if” which means that the hypothesis is true only if the condition is true. For example, If you eat junk food, then you will gain weight is a conditional statement. If you gained weight, then you ate junk food is a converse of a conditional statement. If you do not eat junk food, then you will not gain weight is an inverse of a conditional statement. If yesterday was not Monday, then today is not Tuesday is a contrapositive statement. Today is Tuesday if and only if yesterday was Monday is a bi-conventional statement. Image will be uploaded soon A Conditional Statement Truth Table
In the table above, p→q will be false only if the hypothesis(p) will be true and the conclusion(q) will be false, or else p→q will be true. Conditional Statement ExamplesBelow, you can see some of the conditional statement examples. Example 1) Given, P = I do my work; Q = I get the allowance What does p→q represent? Solution 1) In the sentence above, the hypothesis is “I do my work” and the conclusion is “ I get the allowance”. Therefore, the condition p→q represents the conditional statement, “If I do my work, then I get the allowance”. Example 2) Given, a = The sun is a ball of gas; b = 5 is a prime number. Write a→b in a sentence. Solution 2) The conditional statement a→b here is “if the sun is a ball of gas, then 5 is a prime number”.  FAQs on Conditional Statement1. How many types of conditional statements are there? There are basically 5 types of conditional statements. If statement, if-else statement, nested if-else statement, if-else-if ladder, and switch statement are the basic types of conditional statements. If a function displays a statement or performs a function on the condition if the statement is true. If-else statement executes a block of code if the condition is true but if the condition is false, a new block of code is placed. The switch statement is a selection control mechanism that allows the value of a variable to change the control flow of a program. 2. How are a conditional statement and a loop different from each other? A conditional statement is sometimes used by a loop but a loop is of no use to a conditional statement. A conditional statement is basically a “yes” or a “no” i.e., if yes, then take the first path else take the second one. A loop is more like a cyclic chain starting from the start point i.e., if yes, then take path a, if no, take path b and it comes back to the start point. Conditional statement executes a statement based on a condition without causing any repetition. A loop executes a statement repeatedly. There are two loop variables i.e., for loop and while loop.
How to Understand ‘If-Then’ Conditional Statements: A Comprehensive GuideIn math, and even in everyday life, we often say 'if this, then that.' This is the essence of conditional statements. They set up a condition and then describe what happens if that condition is met. For instance, 'If it rains, then the ground gets wet.' These statements are foundational in math, helping us build logical arguments and solve problems. In this guide, we'll dive into the clear-cut world of conditional statements, breaking them down in both simple terms and their mathematical significance.  Step-by-step Guide: Conditional StatementsDefining Conditional Statements: A conditional statement is a logical statement that has two parts: a hypothesis (the ‘if’ part) and a conclusion (the ‘then’ part). Written symbolically, it takes the form: \( \text{If } p, \text{ then } q \) Where \( p \) is the hypothesis and \( q \) is the conclusion. Truth Values: A conditional statement is either true or false. The only time a conditional statement is false is when the hypothesis is true, but the conclusion is false. Converse, Inverse, and Contrapositive: 1. Converse: The converse of a conditional statement switches the hypothesis and the conclusion. For the statement “If \( p \), then \( q \)”, the converse is “If \( q \), then \( p \)”. 2. Inverse: The inverse of a conditional statement negates both the hypothesis and the conclusion. For the statement “If \( p \), then \( q \)”, the inverse is “If not \( p \), then not \( q \)”. 3. Contrapositive: The contrapositive of a conditional statement switches and negates both the hypothesis and the conclusion. For the statement “If \( p \), then \( q \)”, the contrapositive is “If not \( q \), then not \( p \)”. Example 1: Simple Conditional Statement: “If it is raining, then the ground is wet.” Solution: Hypothesis \(( p )\): It is raining. Conclusion \(( q )\): The ground is wet. Example 2: Determining Truth Value Statement: “If a shape has four sides, then it is a rectangle.” Solution: This statement is false because a shape with four sides could be a square, trapezoid, or other quadrilateral, not necessarily a rectangle. Example 3: Converse, Inverse, and Contrapositive Statement: “If a number is even, then it is divisible by \(2\).” Solution: Converse: If a number is divisible by \(2\), then it is even. Inverse: If a number is not even, then it is not divisible by \(2\). Contrapositive: If a number is not divisible by \(2\), then it is not even. Practice Questions:
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Become a math whiz with AI Tutoring, Practice Questions & more. Conditional StatementsTwo common types of statements found in the study of logic are conditional and biconditional statements. They are formed by combining two statements which we then we call compound statements. What if we were to say, "If it snows, then we don't go outside." This is two statements combined. They are often called if-then statements. As in, "IF it snows, THEN we don't go outside." They are a fundamental building block of computer programming. Writing conditional statementsA statement written in if-then format is a conditional statement. It looks like This represents the conditional statement: "If p then q ." A conditional statement is also called an implication. If a closed shape has three sides, then it is a triangle. The part of the statement that follows the "if" is called the hypothesis. The part of the statement that follows the "then" is the conclusion. So in the above statement, If a closed shape has three sides, (this is the hypothesis) Then it is a triangle. (this is the conclusion) Identify the hypothesis and conclusion of the following conditional statement. A polygon is a hexagon if it has six sides. Hypothesis: The polygon has six sides. Conclusion: It is a hexagon. The hypothesis does not always come first in a conditional statement. You must read it carefully to determine which part of the statement is the hypothesis and which part is the conclusion. Truth table for conditional statementThe truth table for any two given inputs, say A and B , is given by:
Take our conditional statement that if it snows, we do not go outside. If it is snowing ( A is true) and we do go outside ( B is false), then the statement A → B is false. If it is not snowing ( A is false), it doesn't matter if we go outside or not ( B is true or false), because A → B is impossible to determine if A is false, so the statement A → B can still be true. Biconditional statementsA biconditional statement is a combination of a statement and its opposite written in the format of "if and only if." For example, "Two line segments are congruent if and only if they are the same length." This is a combination of two conditional statements. "Two line segments are congruent if they are the same length." "Two line segments are the same length if they are congruent." A biconditional statement is true if and only if both the conditional statements are true. Biconditional statements are represented by the symbol: p ↔ q p ↔ q = p → q ∧ q → p Writing biconditional statementsWrite the two conditional statements that make up this biconditional statement: I am punctual if and only if I am on time to school every day. The two conditional statements that have to be true to make this statement true are:
A rectangle is a square if and only if the adjacent sides are congruent.
Topics related to the Conditional StatementsConjunction Counter Example Biconditional Statement Flashcards covering the Conditional StatementsSymbolic Logic Flashcards Introduction to Proofs Flashcards Practice tests covering the Conditional StatementsIntroduction to Proofs Practice Tests Get help learning about conditional statementsUnderstanding conditional statements can be tricky, especially when it gets deeper into programming language. If your student needs a boost in their comprehension of conditional statements, have them meet with an expert tutor who can give them 1-on-1 support in a setting free from distractions. A tutor can work at your student's pace so that tutoring is efficient while using their learning style - so that tutoring is effective. To learn more about how tutoring can help your student master conditional statements, contact the Educational Directors at Varsity Tutors today.
Converse, Inverse & Contrapositive of Conditional StatementConverse, inverse, and contrapositive of a conditional statement. What we want to achieve in this lesson is to be familiar with the fundamental rules on how to convert or rewrite a conditional statement into its converse, inverse, and contrapositive. But first, we need to review what a conditional statement is because it is the foundation or precursor of the three related sentences that we are going to discuss in this lesson. What is a Conditional Statement?A conditional statement takes the form “If [latex]p[/latex], then [latex]q[/latex]” where [latex]p[/latex] is the hypothesis while [latex]q[/latex] is the conclusion. A conditional statement is also known as an implication . Sometimes you may encounter (from other textbooks or resources) the words “antecedent” for the hypothesis and “consequent” for the conclusion. Don’t worry, they mean the same thing. In addition, the statement “If [latex]p[/latex], then [latex]q[/latex]” is commonly written as the statement “[latex]p[/latex] implies [latex]q[/latex]” which is expressed symbolically as [latex]{\color{blue}p} \to {\color{red}q}[/latex]. Given a conditional statement, we can create related sentences namely: converse , inverse , and contrapositive . They are related sentences because they are all based on the original conditional statement. Let’s go over each one of them! The Converse of a Conditional StatementFor a given conditional statement [latex]{\color{blue}p} \to {\color{red}q}[/latex], we can write the converse statement by interchanging or swapping the roles of the hypothesis and conclusion of the original conditional statement. Therefore, the converse is the implication [latex]{\color{red}q} \to {\color{blue}p}[/latex]. Notice, the hypothesis [latex]\large{\color{blue}p}[/latex] of the conditional statement becomes the conclusion of the converse. On the other hand, the conclusion of the conditional statement [latex]\large{\color{red}p}[/latex] becomes the hypothesis of the converse. The Inverse of a Conditional StatementWhen you’re given a conditional statement [latex]{\color{blue}p} \to {\color{red}q}[/latex], the inverse statement is created by negating both the hypothesis and conclusion of the original conditional statement. Thus, the inverse is the implication ~[latex]\color{blue}p[/latex] [latex]\to[/latex] ~[latex]\color{red}q[/latex]. The symbol ~[latex]\color{blue}p[/latex] is read as “not [latex]p[/latex]” while ~[latex]\color{red}q[/latex] is read as “not [latex]q[/latex]” . The Contrapositive of a Conditional StatementSuppose you have the conditional statement [latex]{\color{blue}p} \to {\color{red}q}[/latex], we compose the contrapositive statement by interchanging the hypothesis and conclusion of the inverse of the same conditional statement. In other words, to find the contrapositive, we first find the inverse of the given conditional statement then swap the roles of the hypothesis and conclusion. Therefore, the contrapositive of the conditional statement [latex]{\color{blue}p} \to {\color{red}q}[/latex] is the implication ~[latex]\color{red}q[/latex] [latex]\to[/latex] ~[latex]\color{blue}p[/latex]. Truth Tables of a Conditional Statement, and its Converse, Inverse, and ContrapositiveNow that we know how to symbolically write the converse, inverse, and contrapositive of a given conditional statement, it is time to state some interesting facts about these logical statements. To save time, I have combined all the truth tables of a conditional statement, and its converse, inverse, and contrapositive into a single table . Here are some of the important findings regarding the table above:
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Conditional Statement In the study of logic, there are two types of statements, conditional statement and bi-conditional statement . These statements are formed by combining two statements, which are called compound statements . Suppose a statement is- if it rains, then we don’t play. This is a combination of two statements. These types of statements are mainly used in computer programming languages such as c, c++, etc. Let us learn more here with examples. Conditional Statement DefinitionA conditional statement is represented in the form of “if…then”. Let p and q are the two statements, then statements p and q can be written as per different conditions, such as;
Points to remember:
Truth Table for Conditional StatementThe truth table for any two inputs, say A and B is given by;
Example: We have a conditional statement If it is raining, we will not play. Let, A: It is raining and B: we will not play. Then;
What is a Bi-Conditional Statement?A statement showing an “if and only if” relation is known as a biconditional statement. An event P will occur if and only if the event Q occurs, which means if P has occurred then it implies Q will occur and vice versa. P: A number is divisible by 2. Q: A number is even. If P will occur then Q will occur and if Q will occur then P will occur. Hence, P will occur if and only if Q will occur. We can say that P↔Q. Conditional Statement ExamplesQ.1: If a > 0 is a positive number, then is a = 10 correct or not? Justify your answer. Solution: Given, a > 0 and is a positive number And it is given a = 10 So the first statement a > 0 is correct because any number greater than 0 is a positive number. But a = 10 is not a correct statement because it can be any number greater than 0. Q.2: Justify P → Q, for the given table below.
Solution: Case 1: We can see, for the first row, in the given table, If statement P is correct, then Q is incorrect and if Q is correct then P is incorrect. Both the statements contradict each other. Hence, P → Q = False Case 2: In the second row of the given table, if P is correct then Q is correct and if Q is correct then P is also correct. Hence, it satisfies the condition. P → Q = True Therefore, we can construct the table;
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 Register with BYJU'S & Download Free PDFsRegister with byju's & watch live videos. Conditional StatementAn "if ... then ..." statement. It has a hypothesis and a conclusion like this: if hypothesis then conclusion Reset password New user? Sign up Existing user? Log in Bayes' Theorem and Conditional ProbabilityAlready have an account? Log in here. Recommended CourseApplied probability. A framework for understanding the world around us, from sports to science.
Bayes' theorem is a formula that describes how to update the probabilities of hypotheses when given evidence. It follows simply from the axioms of conditional probability , but can be used to powerfully reason about a wide range of problems involving belief updates. Given a hypothesis \(H\) and evidence \(E\), Bayes' theorem states that the relationship between the probability of the hypothesis before getting the evidence \(P(H)\) and the probability of the hypothesis after getting the evidence \(P(H \mid E)\) is \[P(H \mid E) = \frac{P(E \mid H)} {P(E)} P(H).\] Many modern machine learning techniques rely on Bayes' theorem. For instance, spam filters use Bayesian updating to determine whether an email is real or spam, given the words in the email. Additionally, many specific techniques in statistics, such as calculating \(p\)-values or interpreting medical results , are best described in terms of how they contribute to updating hypotheses using Bayes' theorem. Explaining Counterintuitive ResultsDeriving bayes' theorem, visualizing bayes’ theorem, diagnosing disease, more examples. Probability problems are notorious for yielding surprising and counterintuitive results. One famous example--or a pair of examples--is the following: A couple has 2 children and the older child is a boy. If the probabilities of having a boy or a girl are both 50%, what's the probability that the couple has two boys? We already know that the older child is a boy. The probability of two boys is equivalent to the probability that the younger child is a boy, which is \(50\%\). A couple has two children, of which at least one is a boy. If the probabilities of having a boy or a girl are both \(50\%\), what is the probability that the couple has two boys? At first glance, this appears to be asking the same question. We might reason as follows: “We know that one is a boy, so the only question is whether the other one is a boy, and the chances of that being the case are \(50\%\). So again, the answer is \(50\%\).” This makes perfect sense. It also happens to be incorrect. Bayes' theorem centers on relating different conditional probabilities . A conditional probability is an expression of how probable one event is given that some other event occurred (a fixed value). For instance, "what is the probability that the sidewalk is wet?" will have a different answer than "what is the probability that the sidewalk is wet given that it rained earlier?" For a joint probability distribution over events \(A\) and \(B\), \(P(A \cap B)\), the conditional probability of \(A\) given \(B\) is defined as \[P(A\mid B) = \frac{P(A\cap B)}{P(B)}.\] In the sidewalk example, where \(A\) is "the sidewalk is wet" and \(B\) is "it rained earlier," this expression reads as "the probability the sidewalk is wet given that it rained earlier is equal to the probability that the sidewalk is wet and it rains over the probability that it rains." Note that \(P(A \cap B)\) is the probability of both \(A\) and \(B\) occurring, which is the same as the probability of \(A\) occurring times the probability that \(B\) occurs given that \(A\) occurred: \(P(B \mid A) \times P(A).\) Using the same reasoning, \(P(A \cap B)\) is also the probability that \(B\) occurs times the probability that \(A\) occurs given that \(B\) occurs: \(P(A \mid B) \times P(B)\). The fact that these two expressions are equal leads to Bayes' Theorem. Expressed mathematically, this is: \[\begin{align} P(A \mid B) &= \frac{P(A\cap B)}{P(B)}, \text{ if } P(B) \neq 0, \\ P(B \mid A) &= \frac{P(B\cap A)}{P(A)}, \text{ if } P(A) \neq 0, \\ \Rightarrow P(A\cap B) &= P(A\mid B)\times P(B)=P(B\mid A)\times P(A), \\ \Rightarrow P(A \mid B) &= \frac{P(B \mid A) \times P(A)} {P(B)}, \text{ if } P(B) \neq 0. \end{align}\] Notice that our result for dependent events and for Bayes’ theorem are both valid when the events are independent. In these instances, \(P(A \mid B) = P(A)\) and \(P(B \mid A) = P(B)\), so the expressions simplify. Bayes' Theorem \[P(A \mid B) = \frac{P(B \mid A)} {P(B)} P(A)\] While this is an equation that applies to any probability distribution over events \(A\) and \(B\), it has a particularly nice interpretation in the case where \(A\) represents a hypothesis \(H\) and \(B\) represents some observed evidence \(E\). In this case, the formula can be written as \[P(H \mid E) = \frac{P(E \mid H)}{P(E)} P(H).\] This relates the probability of the hypothesis before getting the evidence \(P(H)\), to the probability of the hypothesis after getting the evidence, \(P(H \mid E)\). For this reason, \(P(H)\) is called the prior probability , while \(P(H \mid E)\) is called the posterior probability . The factor that relates the two, \(\frac{P(E \mid H)}{P(E)}\), is called the likelihood ratio . Using these terms, Bayes' theorem can be rephrased as "the posterior probability equals the prior probability times the likelihood ratio." If a single card is drawn from a standard deck of playing cards, the probability that the card is a king is 4/52, since there are 4 kings in a standard deck of 52 cards. Rewording this, if \(\text{King}\) is the event "this card is a king," the prior probability \(P(\text{King}) = \frac{4}{52} = \frac{1}{13}.\) If evidence is provided (for instance, someone looks at the card) that the single card is a face card, then the posterior probability \(P(\text{King} \mid \text{Face})\) can be calculated using Bayes' theorem: \[P(\text{King} \mid \text{Face}) = \frac{P(\text{Face} \mid \text{King})}{P(\text{Face})} P(\text{King}).\] Since every King is also a face card, \(P(\text{Face} \mid \text{King}) = 1\). Since there are 3 face cards in each suit (Jack, Queen, King) , the probability of a face card is \(P(\text{Face}) = \frac{3}{13}\). Combining these gives a likelihood ratio of \(\frac{1}{\hspace{2mm} \frac3{13}\hspace{2mm} } = \frac{13}{3}\). Using Bayes' theorem gives \(P(\text{King} \mid \text{Face}) = \frac{13}{3} \frac{1}{13} = \frac{1}{3}\). \(_\square\) You randomly choose a treasure chest to open, and then randomly choose a coin from that treasure chest. If the coin you choose is gold, then what is the probability that you chose chest A? Bayes' theorem clarifies the two-children problem from the first section: 1. A couple has two children, the older of which is a boy. What is the probability that they have two boys? 2. A couple has two children, one of which is a boy. What is the probability that they have two boys? \[\] Define three events, \(A\), \(B\), and \(C\), as follows: \[ \begin{align} A & = \mbox{ both children are boys}\\ B & = \mbox{ the older child is a boy}\\ C & = \mbox{ one of their children is a boy.} \end{align}\] Question 1 is asking for \(P(A \mid B)\), and Question 2 is asking for \(P(A \mid C)\). The first is computed using the simpler version of Bayes’ theorem: \[P(A \mid B) = \frac{P(A)P(B \mid A)}{P(B)} = \frac{ \frac{1}{4}\cdot 1 }{\frac{1}{2}} = \frac{1}{2}.\] To find \(P(A \mid C)\), we must determine \(P(C)\), the prior probability that the couple has at least one boy. This is equal to \(1 - P(\mbox{both children are girls}) = 1 - \frac{1}{4}=\frac{3}{4}\). Therefore the desired probability is \[P(A \mid C) = \frac{P(A)P(C \mid A)}{P(C)} = \frac{\frac{1}{4}\cdot 1}{\frac{3}{4}} = \frac{1}{3}. \ _\square \] For a similarly paradoxical problem, see the Monty Hall problem . Venn diagrams are particularly useful for visualizing Bayes' theorem, since both the diagrams and the theorem are about looking at the intersections of different spaces of events. A disease is present in 5 out of 100 people, and a test that is 90% accurate (meaning that the test produces the correct result in 90% of cases) is administered to 100 people. If one person in the group tests positive, what is the probability that this one person has the disease? The intuitive answer is that the one person is 90% likely to have the disease. But we can visualize this to show that it’s not accurate. First, draw the total population and the 5 people who have the disease: The circle A represents 5 out 100, or 5% of the larger universe of 100 people. Next, overlay a circle to represent the people who get a positive result on the test. We know that 90% of those with the disease will get a positive result, so need to cover 90% of circle A, but we also know that 10% of the population who does not have the disease will get a positive result, so we need to cover 10% of the non-disease carrying population (the total universe of 100 less circle A). Circle B is covering a substantial portion of the total population. It actually covers more area than the total portion of the population with the disease. This is because 14 out of the total population of 100 (90% of the 5 people with the disease + 10% of the 95 people without the disease) will receive a positive result. Even though this is a test with 90% accuracy, this visualization shows that any one patient who tests positive (Circle B) for the disease only has a 32.14% (4.5 in 14) chance of actually having the disease. Main article: Bayesian theory in science and math Bayes’ theorem can show the likelihood of getting false positives in scientific studies. An in-depth look at this can be found in Bayesian theory in science and math . Many medical diagnostic tests are said to be \(X\)% accurate, for instance 99% accurate, referring specifically to the probability that the test result is correct given your condition (or lack thereof). This is not the same as the posterior probability of having the disease given the result of the test. To see this in action, consider the following problem. The world had been harmed by a widespread Z-virus, which already turned 10% of the world's population into zombies. The scientists then invented a test kit with the sensitivity of 90% and specificity of 70%: 90% of the infected people will be tested positive while 70% of the non-infected will be tested negative. If the test kit showed a positive result, what would be the probability that the tested subject was truly zombie? If the solution is in a form of \(\frac{a}{b}\), where \(a\) and \(b\) are coprime positive integers, submit your answer as \(a+b\). A disease test is advertised as being 99% accurate: if you have the disease, you will test positive 99% of the time, and if you don't have the disease, you will test negative 99% of the time. If 1% of all people have this disease and you test positive, what is the probability that you actually have the disease? Balls numbered 1 through 20 are placed in a bag. Three balls are drawn out of the bag without replacement. What is the probability that all the balls have odd numbers on them? In this situation, the events are not independent. There will be a \(\frac{10}{20} = \frac{1}{2}\) chance that any particular ball is odd. However, the probability that all the balls are odd is not \(\frac{1}{8}\). We do have that the probability that the first ball is odd is \(\frac{1}{2}.\) For the second ball, given that the first ball was odd, there are only 9 odd numbered balls that could be drawn from a total of 19 balls, so the probability is \(\frac{9}{19}\). For the third ball, since the first two are both odd, there are 8 odd numbered balls that could be drawn from a total of 18 remaining balls. So the probability is \(\frac{8}{18}\). So the probability that all 3 balls are odd numbered is \(\frac{10}{20} \times \frac{9}{19} \times \frac{8}{18} = \frac{2}{19}.\) Notice that \(\frac{2}{19} \approx 0.105\), whereas \(\frac{1}{8} = 0.125.\) \(_\square\) A family has two children. Given that one of the children is a boy, what is the probability that both children are boys? We assume that the probability of a child being a boy or girl is \(\frac{1}{2}\). We solve this using Bayes’ theorem. We let \(B\) be the event that the family has one child who is a boy. We let \(A\) be the event that both children are boys. We want to find \(P(A \mid B) = \frac{P(B \mid A) \times P(A)}{P(B)}\). We can easily see that \(P(B \mid A) = 1\). We also note that \(P(A) = \frac{1}{4}\) and \(P(B) = \frac{3}{4}\). So \(P(A \mid B) = \frac{1 \times \frac{1}{4}}{\frac{3}{4}} = \frac{1}{3} \). \(_\square\) A family has two children. Given that one of the children is a boy, and that he was born on a Tuesday, what is the probability that both children are boys? Your first instinct to this question might be to answer \(\frac{1}{3}\), since this is obviously the same question as the previous one. Knowing the day of the week a child is born on can’t possibly give you additional information, right? Let’s assume that the probability of being born on a particular day of the week is \(\frac{1}{7}\) and is independent of whether the child is a boy or a girl. We let \(B\) be the event that the family has one child who is a boy born on Tuesday and \(A\) be the event that both children are boys, and apply Bayes’ Theorem. We notice right away that \(P(B \mid A)\) is no longer equal to one. Given that there are 7 days of the week, there are 49 possible combinations for the days of the week the two boys were born on, and 13 of these have a boy who was born on a Tuesday, so \(P( B \mid A) = \frac{13}{49}\). \(P(A)\) remains unchanged at \(\frac{1}{4}\). To calculate \(P(B)\), we note that there are \(14^2\ = 196\) possible ways to select the gender and the day of the week the child was born on. Of these, there are \(13^2 = 169\) ways which do not have a boy born on Tuesday, and \(196 - 169 = 27\) which do, so \(P(B) = \frac{27}{196}\). This gives is that \(P(A \mid B) = \frac{ \frac{13}{49} \times \frac{1}{4}} {\frac{27}{196}} = \frac{13}{27}\). \(_\square\) Note: This answer is certainly not \(\frac{1}{3}\), and is actually much closer to \(\frac{1}{2}\). Zeb's coin box contains 8 fair, standard coins (heads and tails) and 1 coin which has heads on both sides. He selects a coin randomly and flips it 4 times, getting all heads. If he flips this coin again, what is the probability it will be heads? (The answer value will be from 0 to 1, not as a percentage.) There are 10 boxes containing blue and red balls. The number of blue balls in the \(n^\text{th}\) box is given by \(B(n) = 2^n\). The number of red balls in the \(n^\text{th}\) box is given by \(R(n) = 1024 - B(n)\). A box is picked at random, and a ball is chosen randomly from that box. If the ball is blue, and the probability that the \(10^\text{th}\) box was picked can be expressed as \( \frac ab\), where \(a\) and \(b\) are coprime positive integers, find \(a+b\). More probability questions Photo credit: http://www.spi-global.com/  Master concepts like theseLearn more in our Applied Probability course, built by experts for you. Problem Loading... Note Loading... Set Loading... Value Hypothesis Fundamentals: A Complete GuideLast updated on Fri Aug 23 2024 Imagine spending months or even years developing a new feature only to find out it doesn’t resonate with your users, argh! This kind of situation could be any worst Product manager’s nightmare. There's a way to fix this problem called the Value Hypothesis . This idea helps builders to validate whether the ideas they’re working on are worth pursuing and useful to the people they want to sell to. This guide will teach you what you need to know about Value Hypothesis and a step-by-step process on how to create a strong one. At the end of this post, you’ll learn how to create a product that satisfies your users. Are you ready? Let’s get to it! How a Value Hypothesis Helps Product ManagersScrutinizing this hypothesis helps you as a developer to come up with a product that your customers like and love to use. Product managers use the Value Hypothesis as a north star, ensuring focus on client needs and avoiding wasted resources. For more on this, read about the product management process . Definition and Scope of Value HypothesisLet's get into the step-by-step process, but first, we need to understand the basics of the Value Hypothesis: What Is a Value Hypothesis?A Value Hypothesis is like a smart guess you can test to see if your product truly solves a problem for your customers. It’s your way of predicting how well your product will address a particular issue for the people you’re trying to help. You need to know what a Value Hypothesis is, what it covers, and its key parts before you use it. To learn more about finding out what customers need, take a look at our guide on discovering features . The Value Hypothesis does more than just help with the initial launch, it guides the whole development process. This keeps teams focused on what their users care about helping them choose features that their audience will like. Critical Components of a Value Hypothesis A strong Value Hypothesis rests on three key components: Value Proposition: The Value Proposition spells out the main advantage your product gives to customers. It explains the "what" and "why" of your product showing how it eases a particular pain point. This proposition targets a specific group of consumers. To learn more, check out our guide on roadmapping . Customer Segmentation: Knowing and grasping your target audience is essential. This involves studying their demographics, needs, behaviors, and problems. By dividing your market, you can shape your value proposition to address the unique needs of each group. Customer feedback surveys can prove priceless in this process. Find out more about this in our customer feedback surveys guide. Problem Statement : The Problem Statement defines the exact issue your product aims to fix. It should zero in on a real fixable pain point your target users face. For hands-on applications, see our product launch communication plan . Here are some key questions to guide you: What are the primary challenges and obstacles faced by your target users? What existing solutions are available, and where do they fall short? What unmet needs or desires does your target audience have? For a structured approach to prioritizing features based on customer needs, consider using a feature prioritization matrix . Crafting a Strong Value Hypothesis Now that we've covered the basics, let's look at how to build a convincing Value Hypothesis. Here's a two-step method, along with value hypothesis templates, to point you in the right direction: 1. Research and AnalysisTo start with, you need to carry out market research. By carrying out proper market research, you will have an understanding of existing solutions and identify areas in which customers' needs are yet to be met. This is integral to effective idea tracking . Next, use customer interviews, surveys, and support data to understand your target audience's problems and what they want. Check out our list of tools for getting customer feedback to help with this. 2. Finding Out What Customers NeedOnce you've completed your research, it's crucial to identify your customers' needs. By merging insights from market research with direct user feedback, you can pinpoint the key requirements of your customers. Here are some key questions to think about: What are the most significant challenges that your target users encounter daily? Which current solutions are available to them, and how do these solutions fail to fully address their needs? What specific pain points are your target users struggling with that aren't being resolved? Are there any gaps or shortcomings in the existing products or services that your customers use? What unfulfilled needs or desires does your target audience express that aren't currently met by the market? To prioritize features based on customer needs in a structured way, think about using a feature prioritization matrix . Validating the Value HypothesisOnce you've created your Value Hypothesis with a template, you need to check if it holds up. Here's how you can do this: MVP TestingBuild a minimum viable product (MVP)—a basic version of your product with essential functions. This lets you test your value proposition with actual users and get feedback without spending too much. To achieve the best outcomes, look into the best practices for customer feedback software . PrototypingBuild mock-ups to show your product idea. Use these mock-ups to get user input on the user experience and overall value offer. Metrics for EvaluationAfter you've gathered data about your hypothesis, it's time to examine it. Here are some metrics you can use: User Engagement : Monitor stats like time on the platform, feature use, and return visits to see how much users interact with your MVP or mock-up. Conversion Rates : Check conversion rates for key actions like sign-ups, buys, or feature adoption. These numbers help you judge if your value offer clicks with users. To learn more, read our article on SaaS growth benchmarks . Iterative Improvement of Value HypothesisThe Value Hypothesis framework shines because you can keep making it better. Here's how to fine-tune your hypothesis: Set up an ongoing system to gather user data as you develop your product. Look at what users say to spot areas that need work then update your value proposition based on what you learn. Read about managing product updates to keep your hypotheses current. Adaptation to Market ChangesThe market keeps changing, and your Value Hypothesis should too. Stay up to date on what's happening in your industry and watch how users' habits change. Tweak your value proposition to stay useful and ahead of the competition. Here are some ways to keep your Value Hypothesis fresh: Do market research often to keep up with what's happening in your industry and what your competitors are up to. Keep an eye on what users are saying to spot new problems or things they need but don't have yet. Try out different value statements and features to see which ones your audience likes best. To keep your guesses up-to-date, check out our guide on handling product changes . Common Mistakes to AvoidWhile the Value Hypothesis approach is powerful, it's key to steer clear of these common traps: Avoid Confirmation Bias : People tend to focus on data that backs up their initial guesses. But it's key to look at feedback that goes against your ideas and stay open to different views. Watch out for Shiny Object Syndrome : Don't let the newest fads sway you unless they solve a main customer problem. Your value proposition should fix actual issues for your users. Don't Cling to Your First Hypothesis : As the market changes, your value proposition should too. Be ready to shift your hypothesis when new evidence and user feedback comes in. Don't Mix Up Busywork with Real Progress : Getting user feedback is key, but making sense of it brings real value. Look at the data to find useful insights that can shape your product. To learn more about this, check out our guide on handling customer feedback . Value Hypothesis: Action PointsTo build a product that succeeds, you need to know your target users inside out and understand how you help them. The Value Hypothesis framework gives you a step-by-step way to do this. If you follow the steps in this guide, you can create a strong value proposition, check if it works, and keep improving it to ensure your product stays useful and important to your customers. Keep in mind, a good Value Hypothesis changes as your product and market change. When you use data and put customers first, you're on the right track to create a product that works. Want to put the Value Hypothesis framework into action? Check out our top templates for creating product roadmaps to streamline your process. Think about using featureOS to manage customer feedback. This tool makes it easier to collect, examine, and put user feedback to work. 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A conditional statement is a statement that is written in the "If p, then q" format. Here, the statement p is called the hypothesis and q is called the conclusion. It is a fundamental concept in logic and mathematics. Conditional statement symbol: p → q. A conditional statement consists of two parts.
Identify the hypothesis and the conclusion for each of the following conditional statements. (a) If n is a prime number, then n2 has three positive factors. (b) If a is an irrational number and b is an irrational number, then a ⋅ b is an irrational number. (c) If p is a prime number, then p = 2 or p is an odd number.
A conditional statement is an if-then statement connecting a hypothesis (p) and the conclusion (q... 👉 Learn how to label the parts of a conditional statement.
The hypothesis is the first, or "if," part of a conditional statement. The conclusion is the second, or "then," part of a conditional statement. The conclusion is the result of a hypothesis. Figure \(\PageIndex{1}\) If-then statements might not always be written in the "if-then" form. Here are some examples of conditional statements:
Biconditional. A biconditional is a logical conditional statement in which the antecedent and consequent are interchangeable. A biconditional is written as p ↔ q and is translated as " p if and only if q." Because a biconditional statement p ↔ q is equivalent to (p → q) ∧ (q → p), we may think of it as a conditional statement combined ...
A conditional statement has two parts: hypothesis (if) and conclusion (then). In fact, conditional statements are nothing more than "If-Then" statements! Sometimes a picture helps form our hypothesis or conclusion. Therefore, we sometimes use Venn Diagrams to visually represent our findings and aid us in creating conditional statements.
Conditional Statement. A conditional statement is a part of mathematical reasoning which is a critical skill that enables students to analyze a given hypothesis without any reference to a particular context or meaning. In layman words, when a scientific inquiry or statement is examined, the reasoning is not based on an individual's opinion.
A conditional statement, as we've seen, has the form "if p then , q, " and we use the connective . p → q. As many mathematical statements are in the form of a conditional, it is important to keep in mind how to determine if a conditional statement is true or false. A conditional, , p → q, is TRUE if you can show that whenever p is true ...
Definition: A Conditional Statement is…. symbolized by p q, it is an if-then statement in which p is a hypothesis and q is a conclusion. The logical connector in a conditional statement is denoted by the symbol . The conditional is defined to be true unless a true hypothesis leads to a false conclusion. A truth table for p q is shown below. p. q.
A conditional statement is made up of two parts. First, there is a hypothesis that is placed after "if" and before the comma and second is a conclusion that is placed after "then". Here, the hypothesis will be "you do my homework" and the conclusion will be "I will pay you 50 dollars". Now, this statement can either be true or ...
A past hypothetical situation (imaginary, did not happen, or is contrary to fact) influences a present or future hypothetical situation. This is actually a combination of a Class Three Conditional and a Class Two Conditional. Form: If + past perfect tense, would or could + verb stem. Examples:
A biconditional is written as p ↔ q and is translated as " p if and only if q′′. Because a biconditional statement p ↔ q is equivalent to (p → q) ∧ (q → p), we may think of it as a conditional statement combined with its converse: if p, then q and if q, then p. The double-headed arrow shows that the conditional statement goes from ...
A conditional statement is a logical statement that has two parts: a hypothesis (the 'if' part) and a conclusion (the 'then' part). Written symbolically, it takes the form: \( \text{If } p, \text{ then } q \) Where \( p \) is the hypothesis and \( q \) is the conclusion. Truth Values: A conditional statement is either true or false.
A statement written in if-then format is a conditional statement. It looks like. p → q. This represents the conditional statement: "If p then q." A conditional statement is also called an implication. Example 1. If a closed shape has three sides, then it is a triangle. The part of the statement that follows the "if" is called the hypothesis.
Conditional statement: A conditional statement states that if a hypothesis holds, then a conclusion holds. The hypothesis is typically symbolized by p , and the conclusion is typically symbolized ...
Hypothesis and Conclusion of a Conditional Statement. Conditional statements can be symbolized in order to make it easier to manipulate them in logical analysis. Often, an arrow symbol pointing to ...
The Contrapositive of a Conditional Statement. Suppose you have the conditional statement [latex]{\color{blue}p} \to {\color{red}q}[/latex], we compose the contrapositive statement by interchanging the hypothesis and conclusion of the inverse of the same conditional statement.. In other words, to find the contrapositive, we first find the inverse of the given conditional statement then swap ...
A conditional statement is also called implication. The sign of the logical connector conditional statement is →. Example P → Q pronouns as P implies Q. The state P → Q is false if the P is true and Q is false otherwise P → Q is true. Truth Table for Conditional Statement. The truth table for any two inputs, say A and B is given by;
Study with Quizlet and memorize flashcards containing terms like What is the hypothesis of the given statement? If money grows in trees, then you can be rich., Which statement is the converse of the given statement? If you make an insurance claim, then your rates will go up., Which statement is the contrapositive of the given statement? If a person is a banjo player, then the person is a ...
Exercise 46: Use a class one conditional sentence in the past situations that were real and did happen, please. Examples: If I ate too much candy when I was a boy, I threw up. If I had to go shopping with my sister and my mother when I was a boy, I wanted to go home right away. 1.
An "if ... then ..." statement. It has a hypothesis and a conclusion like this: if hypothesis then conclusion
Bayes' theorem is a formula that describes how to update the probabilities of hypotheses when given evidence. It follows simply from the axioms of conditional probability, but can be used to powerfully reason about a wide range of problems involving belief updates.. Given a hypothesis \(H\) and evidence \(E\), Bayes' theorem states that the relationship between the probability of the ...
Biconditional. A biconditional is a logical conditional statement in which the hypothesis and conclusion are interchangeable. A biconditional is written as p ↔ q p ↔ q and is translated as " p p if and only if q′′ q ′ ′. Because a biconditional statement p ↔ q p ↔ q is equivalent to (p → q) ∧ (q → p), ( p → q) ∧ ( q ...
Scrutinizing this hypothesis helps you as a developer to come up with a product that your customers like and love to use. Product managers use the Value Hypothesis as a north star, ensuring focus on client needs and avoiding wasted resources. For more on this, read about the product management process. Definition and Scope of Value Hypothesis