Question

Find the contrapositive of the statement 'If two numbers are not equal, then their squares are not equal'.
(a) If the squares of two numbers are not equal, then the numbers are not equal
(b) If the squares of two numbers are not equal, then the numbers are equal
(c) If the squares of two numbers are equal, then the numbers are equal
(d) If the squares of two numbers are equal, then the numbers are not equal

Answer 1

In this question, in order to find contrapositive of the statement 'If two numbers are not equal, then their squares are not equal' we will first have to determine which statement implies which statement. We using use the property that if the statement $X$ implies statement $Y$, which is denoted by $X\Rightarrow Y$, then the contrapositive statement of $X\Rightarrow Y$ is given by $\sim X\Rightarrow \sim Y$ which simply means that if statement $X$ is not true implies statement $Y$is not true.
The notation $\sim$ simply means the negation.
Now this we will determine the contrapositive of the statement 'If two numbers are not equal, then their squares are not equal'.

Complete step-by-step answer :
Let us suppose that the statement $p$ is given by “Two numbers are not equal”.
Also let us suppose that the statement $q$ is given by “Sum of these two numbers are not equal”.
We are given that 'If two numbers are not equal, then their squares are not equal’.
Which means that statement $p$ implies statement $q$.
That is $p\Rightarrow q$.
Now since we know that if the statement $X$ implies statement $Y$, which is denoted by $X\Rightarrow Y$, then the contrapositive statement of $X\Rightarrow Y$ is given by $\sim X\Rightarrow \sim Y$ which simply means that if statement $X$ is not true implies statement $Y$is not true.
The notation $\sim$ simply means the negation.
Thus we have that the contrapositive statement of $p\Rightarrow q$ is given by $\sim p\Rightarrow \sim q$.
Now negation of the statement $p$ denoted by $\sim p$ is given by
“Two numbers are equal”.
Also now negation of the statement $q$ denotes by $\sim q$,is given by
“Sum of the two numbers are equal”.
Therefore the statement $\sim p\Rightarrow \sim q$ actually means that “if the sum of the squares of two numbers are equal, then the two numbers are equals”.
Therefore we have that the contrapositive of the statement 'If two numbers are not equal, then their squares are not equal' is given by
“if the sum of the squares of two numbers are equal, then the two numbers are equals”.
Hence option (c) is correct.

Note : In this problem, we are using use the property that if the statement $X$ implies statement $Y$, which is denoted by $X\Rightarrow Y$, then the contrapositive statement of $X\Rightarrow Y$ is given by $\sim X\Rightarrow \sim Y$ which simply means that if statement $X$ is not true implies statement $Y$is not true. Where the notation $\sim$ simply means the negation.