Introduction to Proofs

What Is a Mathematical Proof?

A mathematical proof is a logical argument that establishes the truth of a mathematical statement beyond any doubt. Unlike empirical sciences, where evidence supports theories, proofs in mathematics provide absolute certainty.

Every proof starts with axioms (assumed truths) and uses rules of inference to reach conclusions.

Terminology

Theorem    - A statement that has been proved
Lemma      - A smaller result used to prove a theorem
Corollary  - A result that follows directly from a theorem
Axiom      - A statement assumed to be true without proof
Conjecture - A statement believed to be true but not yet proved

Structure of a Proof

A proof typically follows this structure:

1. State what you are proving
2. Clearly state any assumptions
3. Use logical reasoning step by step
4. Arrive at the conclusion
5. Mark the end (often with Q.E.D. or a box)

Example: Proving an Even Number Property

Theorem: The sum of two even integers is even.

Proof:
Let m = 2a and n = 2b where a, b are integers.
Then m + n = 2a + 2b = 2(a + b).
Since a + b is an integer, 2(a + b) is even.
Therefore, the sum of two even integers is even.  ∎