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Contrapositive of "If n^3-1 is even then...

Contrapositive of "If `n^3-1` is even then n is odd" is-

A

If n is odd then `n^3-1` is not even

B

If n is not odd then `n^3-1` is even

C

If n is not odd then `n^3-1` is not even

D

If n is odd then `n^3-1` is even

Text Solution

AI Generated Solution

The correct Answer is:
To find the contrapositive of the statement "If \( n^3 - 1 \) is even, then \( n \) is odd", we can follow these steps: ### Step 1: Identify the components of the statement The statement can be expressed in the form "If \( p \), then \( q \)", where: - \( p \): \( n^3 - 1 \) is even - \( q \): \( n \) is odd ### Step 2: Write the negations of \( p \) and \( q \) To find the contrapositive, we need to negate both \( q \) and \( p \): - Negation of \( q \) (not \( q \)): \( n \) is not odd (which means \( n \) is even) - Negation of \( p \) (not \( p \)): \( n^3 - 1 \) is not even (which means \( n^3 - 1 \) is odd) ### Step 3: Formulate the contrapositive The contrapositive of the statement "If \( p \), then \( q \)" is "If not \( q \), then not \( p \)". Therefore, we can write: - If \( n \) is even (not \( q \)), then \( n^3 - 1 \) is odd (not \( p \)). ### Step 4: Write the final contrapositive statement Thus, the contrapositive statement can be expressed as: - "If \( n \) is even, then \( n^3 - 1 \) is odd." ### Final Answer The contrapositive of "If \( n^3 - 1 \) is even, then \( n \) is odd" is: - "If \( n \) is even, then \( n^3 - 1 \) is odd." ---
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