Question

Name the three different types of proofs you saw in this lesson. Give a description of each.

Answer 1

Here are three common types of proofs (with short descriptions and when to use each):

1. Direct proof
- Description: Start from the given assumptions and use definitions, algebraic manipulation, and known results to arrive directly at the statement you want to prove.
- When to use: Best for straightforward implications where you can transform premises step by step into the conclusion.
- Quick example idea: To prove “if n is even then n^2 is even,” write n = 2k and show n^2 = 4k^2 = 2(2k^2).

2. Proof by contrapositive
- Description: To prove an implication “if P then Q,” prove the logically equivalent statement “if not Q then not P.” Often the contrapositive is easier to work with.
- When to use: Useful when negating the conclusion gives a simpler route back to negating the premise.
- Quick example idea: To prove “if n^2 is even then n is even,” assume n is not even (odd) and show n^2 would be odd, contradicting n^2 even; equivalently one proves the contrapositive.

3. Proof by contradiction (indirect proof)
- Description: Assume the statement you want to prove is false, then deduce a logical contradiction (something impossible, like A and not-A or a violation of a known fact). Conclude the assumption was false, so the statement is true.
- When to use: Good for existence statements or statements where assuming the negation forces an impossible chain of consequences.
- Quick example idea: Prove √2 is irrational by assuming √2 is rational (equal to a/b in lowest terms) and showing that both a and b must be even, contradicting lowest-terms assumption.

If you meant a different set of three (for example, including mathematical induction), tell me and I’ll list those instead.