Appendix 1 — Proofs in Mathematics

The Purpose of Proofs

A proof transforms a guess or a pattern into an unshakable truth. It is a logical chain of reasoning that starts from accepted statements (axioms, definitions, or previously proven theorems) and, through valid steps, arrives at a new statement (the theorem).

Key Types of Proofs Covered

Three fundamental proof techniques, each a different tool for establishing truth.

1. Direct Proof

Assume the hypothesis (the "if" part) is true, then use logical deductions to show the conclusion (the "then" part) must also be true.

Structure: If $P$ is true, then $Q$ is true.

Example: Prove that the product of two even integers is even.

• Let $a$ and $b$ be even integers, so $a = 2m$ and $b = 2n$ for some integers $m, n$.
• Their product is $a \times b = (2m)(2n) = 4mn = 2(2mn)$.
• Since $2mn$ is an integer, $a \times b$ is of the form $2 \times (\text{integer})$, hence even.

2. Proof by Contradiction

Assume the opposite of what you want to prove (the conclusion is false) and show this leads to a contradiction. Since the assumption is impossible, the original statement must be true.

Structure: To prove $P \Rightarrow Q$, assume $P$ true and $Q$ false, derive a contradiction; therefore $Q$ must be true.

Example: Prove that $\sqrt{2}$ is irrational. …