Question

For which of these would a proof by contraposition be a better approach than a direct approach?

A. If a is odd and b Is odd, then ab is odd.
B. If 3n+2 is odd, then n is odd.
C. If n is odd, then n3 + 1 is even
D. Prove that if 1/x is irrational then x is irrational

Answer 1

A proof by contraposition is best applied to the statement 'If 3n+2 is odd, then n is odd' since it deals with odd and even numbers, allowing for simpler logic when dealing with divisibility by 2.

Step-by-step explanation:

Proof by contraposition can be a powerful method when the direct approach is less straightforward. When faced with the statement, If 3n+2 is odd, then n is odd (Option B), using a contrapositive approach can simplify the process. To prove this by contraposition, we assume the negation of the conclusion, stating that n is even, and demonstrate that this leads to 3n+2 being even, which is the negation of the original hypothesis. This method is easier than proving the original statement directly, because dealing with the oddness or evenness of numbers often involves looking at their divisibility by 2, which is more straightforwardly handled in the contrapositive form.