Different types of statements are used in mathematics to convey certain theorems, corollaries, or prove some ideas. One such statement is the converse statement. This Buzzle article explains how to write one, along with some examples of converse statements.
Converse vs. Inverse
In inverse statements, the opposite of the original hypothesis and conclusion is written, whereas in a converse statement, only the hypothesis and the conclusion is exchanged. The meaning of the statement does not change in an inverse statement.
A very important type of statement, the converse statement is mostly used in geometrical theorems. Understanding or writing a converse theorem is not very difficult.
In this Buzzle write-up, we discuss the meaning of a converse statement, how it is written, and some examples.
Converse statements are a type of conditional statements. So let us understand in brief what conditional statements are. They are basically if and then statements.
Here’s an example: If it rains, then I won’t go to school. The first part, i.e., if it rains is called the hypothesis, whereas the latter part then I won’t go to school is called conclusion. When one condition is fulfilled, then the other condition can happen. Conditional statements are used in mathematical theorems. Sometimes, there can be two or more conditions as well.
What is a Converse Statement?
Converse statement is a statement in which the hypothesis and conclusion is interchanged. For example, statement: If the angle is less than 90º, then it is an acute angle.
Converse: If the angle is acute, it is less than 90º.
Here you can see that the hypothesis of the statement becomes the conclusion in the converse, and the conclusion becomes hypothesis.
The most important factor to be considered is that the converse statement may not be true in all cases.
For e.g.: If you are a girl, then you are a human.
Converse of this statement will be: If you are a human, then you are a girl.
It is obviously not true.
In a converse statement, the hypothesis is called ‘p’, and the conclusion is called ‘q’. The symbol for a converse statement is as follows:
Statement: If p, then q. (p ➔ q)
Converse: If q, then p. (q ➔ p)
The truth table will help us understand the concept of converse statements easily. Let p be the hypothesis, and q be the conclusion. We will discuss the different cases using a truth table.
|P||Q||P → Q||Q → P (converse)|
When the hypothesis and the conclusion are true, both statement and its converse are true. When the hypothesis is true, but the conclusion is false, then the statement is false, but the converse is true. If the hypothesis is false and the conclusion is true, then the statement is true and the converse is false. Finally, if both hypothesis and conclusion are false, then both statement and its converse are true.
Examples of Converse Statements
When converse is true
Statement: If a number ends in 0, then it is a multiple of 10.
Converse: If a number is a multiple of 10, then it ends in 0.
Statement: If the measure of an angle is less than 90, then it is an acute angle.
Converse: If the angle is acute, then its measure is less than 90.
Statement: If the number has only two divisors, then it is prime.
Converse: If the number is prime, then it has only two divisors.
Statement: If two lines don’t intersect, they are parallel.
Converse: If two lines are parallel then they don’t intersect.
Statement: If the number is divisible by 2, then it is an even number.
Converse: If it is an even number, then it is divisible by 2.
When converse is not true
Statement: If two angles are vertically opposite, then they are congruent.
Converse: If two angles are congruent, then they are vertically opposite.
Statement: If two numbers are even, then their sum will be even.
Converse: If the sum of two numbers is even, then they are even.
Statement: If a triangle is equilateral, then it is isosceles.
Converse: If a triangle is isosceles, then it is equilateral.
Statement: If a quadrilateral is square, then it is a rectangle.
Converse: If a quadrilateral is rectangle, then it is a square.
Statement: If you study, then you will pass the test.
Converse: If you pass the test, then you will study.