Question

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

Answer 1

Common introductory proof types are:

1. Direct proof
- Description: Start from the assumptions (the "if" part) and use definitions, algebra, known theorems, and logical steps to arrive at the conclusion (the "then" part).
- Example idea: To prove "the sum of two even integers is even," write each as 2k and 2m, add them to get 2(k+m), which is even.

2. Proof by contrapositive
- Description: To prove an implication "if P then Q," prove its logically equivalent contrapositive "if not Q then not P." This is useful when the contrapositive is easier to establish directly.
- Example idea: To show "if n^2 is even then n is even," prove the contrapositive: if n is odd then n^2 is odd.

3. Proof by contradiction (indirect proof)
- Description: Assume the statement you want to prove is false (often assume P and not Q, or assume a proposition and its negation), then deduce a logical contradiction. Concluding a contradiction lets you infer the original statement is true.
- Example idea: The classic proof that sqrt(2) is irrational assumes sqrt(2) = a/b in lowest terms, derives a contradiction about parity, and therefore concludes no such fraction exists.

If you meant a different lesson (for example one that emphasized induction or proof by exhaustion), tell me and I’ll adapt the list.