Proof / Direct And Contrapositive
Least You Need to Know: Direct Proof and Contrapositive
When a statement has the form **if P, then Q**, you need to choose a proof path that preserves logic instead of guessing from examples.
The least you need to know
- A direct proof starts by assuming P and showing Q.
- A contrapositive proof shows that not Q implies not P.
- A few examples do not prove a universal statement.
- One counterexample is enough to disprove a universal statement.
Key notation
P → Q
if P then Q
¬Q → ¬P
contrapositive
∴
therefore
∈
is an element of
Tiny worked example
- Claim: If n is even, then n^2 is even.\n- Let n = 2k. Then n^2 = 4k^2 = 2(2k^2), so n^2 is even.\n- The proof starts from the hypothesis and rewrites it in a useful form.
Common mistakes
- Students often try to prove an implication by testing one or two examples.
- Students often confuse converse with contrapositive.
- Students often start from the conclusion instead of the hypothesis.
How to recognize this kind of problem
- If the claim says 'if ... then ...', identify hypothesis and conclusion before writing anything.
- Use contrapositive when not Q is easier to analyze than P.
- Check whether the task asks for proof or disproof.