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Question

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A. If the squares of the 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 not equal.

D. None of these

Answer

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Hint: In logic and mathematics, the contrapositive of a conditional statement of the form “If \[p\] then \[q\]” is “If \[ \sim q\] then \[ \sim p\]”. Symbolically, the contrapositive of \[p \to q\] is \[ \sim q \to \sim p\]. So, take the first statement as \[p\] and the second statement as \[q\] and find the contrapositive as stated above. So, use this concept to reach the solution of the given problem.

Complete step-by-step answer:

Statement: “If two numbers are not equal, then their squares are not equal”.

Let the first statement i.e., If two numbers are not equal is \[p\]

And the second statement i.e., their squares are not equal is \[q\]

Now the total statement is given by \[p \to q\]

We know that the contrapositive statement of \[p \to q\] is \[ \sim q \to \sim p\].

So, consider

The negative statement of \[q\] i.e., \[ \sim q\] is If the squares of two numbers are equal

The negative statement of \[p\] i.e., \[ \sim p\] is the two numbers are equal

Hence the contrapositive statement \[ \sim q \to \sim p\] is given by “If the squares of the two numbers are equal, then the numbers are equal”.

Thus, the correct option is A. If the squares of the two numbers are equal, then the numbers are equal.

Note: Contraposition refers to the inference of going from a conditional statement into its logically equivalent contrapositive, and an associated proof method known as proof by contraposition. The contrapositive of a statement has its antecedent and consequent inverted and flipped.

Complete step-by-step answer:

Statement: “If two numbers are not equal, then their squares are not equal”.

Let the first statement i.e., If two numbers are not equal is \[p\]

And the second statement i.e., their squares are not equal is \[q\]

Now the total statement is given by \[p \to q\]

We know that the contrapositive statement of \[p \to q\] is \[ \sim q \to \sim p\].

So, consider

The negative statement of \[q\] i.e., \[ \sim q\] is If the squares of two numbers are equal

The negative statement of \[p\] i.e., \[ \sim p\] is the two numbers are equal

Hence the contrapositive statement \[ \sim q \to \sim p\] is given by “If the squares of the two numbers are equal, then the numbers are equal”.

Thus, the correct option is A. If the squares of the two numbers are equal, then the numbers are equal.

Note: Contraposition refers to the inference of going from a conditional statement into its logically equivalent contrapositive, and an associated proof method known as proof by contraposition. The contrapositive of a statement has its antecedent and consequent inverted and flipped.