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.
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
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 it
- 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.
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