A mathematical proof is a clear argument that shows a statement must be true, not just that it seems true from examples. Proof methods are tools for organizing reasoning so each step follows from definitions, known facts, or earlier results. Learning these methods matters because proof is the language used to justify results in algebra, geometry, calculus, number theory, and computer science.
Good proofs also build habits of precise thinking and careful communication.
Different proof techniques fit different kinds of statements. A direct proof starts with the assumptions and moves step by step to the conclusion, while a proof by contrapositive proves an equivalent statement instead. A proof by contradiction assumes the statement is false and shows that this leads to an impossibility.
Counterexamples are used to disprove universal claims by finding just one case where the claim fails.
Key Facts
- Direct proof pattern: Assume P is true, use definitions and theorems, then prove Q is true.
- Contrapositive equivalence: P implies Q is logically equivalent to not Q implies not P.
- Contradiction pattern: To prove P, assume not P and derive a contradiction such as A and not A.
- Counterexample rule: A universal statement is false if one example satisfies the hypothesis but not the conclusion.
- Even integer definition: n is even means n = 2k for some integer k.
- Odd integer definition: n is odd means n = 2k + 1 for some integer k.
Vocabulary
- Proof
- A proof is a logical argument that establishes a mathematical statement using definitions, accepted facts, and valid reasoning.
- Direct proof
- A direct proof shows P implies Q by starting with P and deriving Q through valid steps.
- Contrapositive
- The contrapositive of P implies Q is not Q implies not P, and it always has the same truth value as the original statement.
- Contradiction
- A contradiction is a statement that cannot be true, such as saying a number is both even and not even.
- Counterexample
- A counterexample is a single example that proves a universal claim is false.
Common Mistakes to Avoid
- Testing only a few examples and calling it a proof is wrong because examples can support a pattern but cannot prove it for all cases.
- Proving the converse instead of the original statement is wrong because Q implies P is not logically equivalent to P implies Q.
- Using contradiction without clearly stating the negation is wrong because the reader must know exactly what assumption leads to the impossibility.
- Giving a counterexample that does not satisfy the hypothesis is wrong because it cannot disprove the claimed implication.
Practice Questions
- 1 Use a direct proof to show that if n is an even integer, then n^2 is even.
- 2 Use proof by contrapositive to prove that if n^2 is odd, then n is odd.
- 3 A student claims that if a number is divisible by 6, then it is divisible by 12. Explain whether this statement is true or false, and justify your answer using an appropriate proof method.