Implications and contrapositive

• An implication has the form P -> Q, read "if P then Q." P is the hypothesis and Q is the conclusion. The most common beginner mistake is confusing an implication with its converse. "If it's a square, then it's a rectangle" is true, but "if it's a rectangle, then it's a square" is false. In proofs, you should treat an implication as a promise: you are allowed to assume P, and your job is to logically derive Q. Also note that P -> Q is considered true whenever P is false, which is why we focus on situations where P actually holds.
• A key technique is proving an implication by proving its contrapositive because the contrapositive is logically equivalent and often easier. The contrapositive of P -> Q is not Q -> not P. Example: proving "if n^2 is even then n is even" is tricky directly, but the contrapositive "if n is odd then n^2 is odd" is straightforward using n = 2k + 1. In practice, when the conclusion is a "simple" property (like evenness) and the hypothesis is "complicated" (like n^2 is even), contrapositive is often the cleanest route.

• If n^2 is even then n is even (proved by contrapositive: odd n gives odd n^2).

• 1Write the contrapositive: If a number is divisible by 4, then it is even.
• 2Is the converse true? If a number is even then divisible by 4?
• 3Prove: If n is divisible by 6, then n is divisible by 3.

MCQ

What is the contrapositive of 'If a number is divisible by 4, then it is even'?