Converse Statement is a type of conditional statement where the hypothesis (or antecedent) and conclusion (or consequence) are reversed relative to a given conditional statement.
For instance, consider the statement: “If a triangle ABC is an equilateral triangle, then all its interior angles are equal.” The converse of this statement would be: “If all the interior angles of triangle ABC are equal, then it is an equilateral triangle”
In this article, we will discuss all the things related to the Converse statement in detail.
Table of Content
What is a Converse Statement?
A converse statement is a proposition formed by interchanging the hypothesis and conclusion of a conditional statement.
In simpler terms, it's like flipping the order of "if" and "then" in a statement. For example, in the conditional statement "If it is raining, then the ground is wet", the converse statement would be "If the ground is wet, then it is raining."
P→Q is Q→P
Note: The truth of the original statement doesn't necessarily imply the truth of its converse, and vice versa.
Definition of Converse Statement
A converse statement is formed by exchanging the hypothesis and conclusion of a conditional statement while retaining the same meaning.
For instance, if the original statement is "If A, then B," the converse is "If B, then A." The validity of a converse statement doesn't guarantee the truth of the original statement, and vice versa.
How to Write a Converse Statement?
To write a converse statement, you simply switch the hypothesis and conclusion of a conditional statement while maintaining the same meaning. For example, if the original statement is "If it is raining (hypothesis), then the ground is wet (conclusion)," the converse statement would be "If the ground is wet (hypothesis), then it is raining (conclusion)." Remember, the converse statement may not always be true, even if the original statement is.
Examples of Converse Statements
Some examples of converse statements are:
- Original Statement: If a shape is a square, then it has four equal sides.
Converse Statement: If a shape has four equal sides, then it is a square.
- Original Statement: If it is summer, then the weather is hot.
Converse Statement: If the weather is hot, then it is summer.
- Original Statement: If a number is divisible by 2, then it is even.
Converse Statement: If a number is even, then it is divisible by 2.
- Original Statement: If a person is a teenager, then they are between 13 and 19 years old.
Converse Statement: If a person is between 13 and 19 years old, then they are a teenager.
- Original Statement: If an animal is a dog, then it has fur.
Converse Statement: If an animal has fur, then it is a dog.
Examples of Converse Statements in Mathematics or Logic
Some examples of converse statements in mathematics or logic:
- Original Statement: If two angles are congruent, then they have the same measure.
Converse Statement: If two angles have the same measure, then they are congruent.
- Original Statement: If a number is divisible by 6, then it is divisible by 2 and 3.
Converse Statement: If a number is divisible by 2 and 3, then it is divisible by 6.
- Original Statement: If two lines are perpendicular, then their slopes are negative reciprocals of each other.
Converse Statement: If the slopes of two lines are negative reciprocals of each other, then the lines are perpendicular.
Converse, Inverse and Contrapositive Statements
Inverse Statement: The inverse of a conditional statement is formed by negating both the hypothesis and the conclusion of the original statement.
Contrapositive Statement: The contrapositive of a conditional statement is formed by switching the hypothesis and conclusion of the original statement and negating both.
| Statement | Converse | Inverse | Contrapositive |
|---|---|---|---|
| If p, then q | If q, then p | If not p, then not q | If not q, then not p |
Example of Inverse Statements
Original Statement: If a number is even, then it is divisible by 2.
Inverse Statement: If a number is not even, then it is not divisible by 2.
Original Statement: If x > 5, then 2x > 10.
Inverse Statement: If x ≤ 5, then 2x ≤ 10.
Example of Contrapositive Statements
Original Statement: If a shape is a square, then it has four equal sides.
Contrapositive Statement: If a shape does not have four equal sides, then it is not a square.
Original Statement: If a number is even, then it is divisible by 2.
Contrapositive: If a number is not divisible by 2, then it is not even.
Truth Table for Converse Statement
To create a truth table for the converse statement, we need to consider both the original statement and its converse.
Let's represent the original statement as "If p, then q" or "p → q" where p is the hypothesis and q is the conclusion. The converse of this statement is "If q, then p" or "q → p". Then truth table is given by:
| Original | Converse | ||||
|---|---|---|---|---|---|
| p | q | ~p | ~q | p → q | q → p |
| TRUE | TRUE | FALSE | FALSE | TRUE | TRUE |
| TRUE | FALSE | FALSE | TRUE | FALSE | TRUE |
| FALSE | TRUE | TRUE | FALSE | TRUE | FALSE |
| FALSE | FALSE | TRUE | TRUE | TRUE | TRUE |
Truth Table for Inverse and Contrapositive Statement
To create a truth table for the inverse and contrapositive statements, let's start with the original statement "If p, then q" or "p → q" where p is the hypothesis and q is the conclusion. The inverse of this statement is "If not p, then not q" or "~p → ~q", and the contrapositive is "If not q, then not p" or "~q → ~p". Then truth table is given by:
| Original | Inverse | Contrapositive | ||||
|---|---|---|---|---|---|---|
| p | q | ~p | ~q | p → q | ~p → ~q | ~q → ~p |
| TRUE | TRUE | FALSE | FALSE | TRUE | TRUE | TRUE |
| TRUE | FALSE | FALSE | TRUE | FALSE | TRUE | FALSE |
| FALSE | TRUE | TRUE | FALSE | TRUE | FALSE | TRUE |
| FALSE | FALSE | TRUE | TRUE | TRUE | TRUE | TRUE |
Solved Questions on Converse Statement
Example 1: If all squares are rectangles, are all rectangles squares?
Converse: If a shape is a rectangle, then it is a square.
Solution:
The original statement says that all squares are rectangles. This is true because a square, by definition, has four sides of equal length and four right angles, making it a special type of rectangle where all sides are equal. However, the converse statement is not necessarily true. Not all rectangles are squares because rectangles can have unequal side lengths, whereas squares have all sides equal. Therefore, the converse statement is false.
Example 2: If all right angles are 90 degrees, are all 90 degree angles right angles?
Converse: If an angle measures 90 degrees, then it is a right angle.
Solution:
The original statement is true because a right angle, by definition, measures 90 degrees. However, the converse statement is also true. If an angle measures 90 degrees, then it must be a right angle, as any angle measuring exactly 90 degrees forms a perfect right angle.
Example 3: If a number is divisible by 3, then it is an odd number.
Converse: If a number is an odd number, then it is divisible by 3.
Solution:
The original statement is false. While it is true that all odd numbers are not divisible by 2, they are not necessarily divisible by 3. For example, the number 5 is an odd number but is not divisible by 3. Therefore, the converse statement is also false because not all odd numbers are divisible by 3.
Example 4: If a shape has four sides, then it is a quadrilateral.
Converse: If a shape is a quadrilateral, then it has four sides.
Solution:
The original statement is true. A quadrilateral is defined as a polygon with four sides, so any shape with four sides is indeed a quadrilateral. Similarly, the converse statement is true. If a shape is a quadrilateral, then it must have four sides because that is a defining characteristic of a quadrilateral. Therefore, both the original statement and its converse are true.
Converse Statement: Practice Questions
Q1: If all birds have wings, do all winged creatures have beaks?
Q2: If all triangles have three sides, do all polygons with three sides have to be triangles?
Q3: If all vehicles are cars, are all cars vehicles?