Question

Contrapositive of the statement
“If two numbers are not equal, then their squares are not equal.” Is
A. If the squares of two numbers are equal, then the numbers are equal.
B. If the squares of two numbers are equal, then the numbers are not equal.
C. If the squares of two numbers are not equal, then the numbers are equal.
D. If the squares of two numbers are not equal, then the numbers are not equal.

Answer 1

First we split the given statement into two separate meaningful statements. Let us assume the first statement to be $p$ and the second statement to be $q$. Now, we use the identity which states that if $p$ implies $q$ i.e. $p\Rightarrow q$ then its contrapositive statement will be negation of $q$ implies negation of $p$ i.e. $\sim q\Rightarrow \sim p$.

Complete step by step answer:
We have been given a statement “If two numbers are not equal, then their squares are not equal.”
We have to find the contrapositive of the given statement.
Let us first split the given statement into two separate meaningful statements.
Let the first statement “if two numbers are not equal” $=p$
Let the second statement “their squares are not equal” $=q$
Now, we know that the contrapositive of a statement $p\Rightarrow q$ is given by $\sim q\Rightarrow \sim p$
Now, we have to find the negation of both the statements.
First consider statement
$p=$ if two numbers are not equal
So, the negation of statement will be $\sim p=$ if two numbers are equal.
Now, consider statement $q=$ their squares are not equal
So, the negation of the statement will be $\sim q=$ their squares are equal.
So, the contrapositive of statements will be $\sim q\Rightarrow \sim p$
If the squares of two numbers are equal, then the numbers are equal.

So, the correct answer is “Option A”.

Note: Contrapositive statements are formed by contradicting the hypothesis and conclusion of a given preposition and then interchange them. The possibility of mistakes while solving such types of questions is that students can write the statement $p$ first in the contrapositive, which gives the incorrect answer. Always write the negation of statement $q$ first and then negation of statement $p$ i.e. $\sim q\Rightarrow \sim p$.