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
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:
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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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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How to Understand ‘If-Then’ Conditional Statements: A Comprehensive Guide
In 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 Statements
Defining 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:
- Write the converse, inverse, and contrapositive for the statement: “If a bird is a penguin, then it cannot fly.”
- Determine the truth value of the statement: “If a shape has three sides, then it is a triangle.”
- For the statement “If an animal is a cat, then it is a mammal,” which of the following is its converse? a) If an animal is a mammal, then it is a cat. b) If an animal is not a cat, then it is not a mammal. c) If an animal is not a mammal, then it is not a cat.
- Converse: If a bird cannot fly, then it is a penguin. Inverse: If a bird is not a penguin, then it can fly. Contrapositive: If a bird can fly, then it is not a penguin.
- The statement is true. A shape with three sides is defined as a triangle.
- a) If an animal is a mammal, then it is a cat.
by: Effortless Math Team about 1 year ago (category: Articles )
Effortless Math Team
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