Mathematical proof is the mechanism by which mathematics advances: every theorem rests on a chain of deductive reasoning from axioms through established results to a new conclusion. Knowing which technique to reach for — direct proof, contradiction, induction, well-ordering, or a combinatorial argument — is itself a core mathematical skill. The techniques covered here range from foundational (direct proof and contrapositive) through structural (induction and its variants) to powerful specialized tools (the probabilistic method and bijective proof). A unifying insight is that several pairs of techniques are logically equivalent — induction and well-ordering, contrapositive and direct proof of the negation — so the choice is always about which path produces the clearest argument.
What This Cheat Sheet Covers
This topic spans 13 focused tables and 89 indexed concepts. Below is a complete table-by-table outline of this topic, spanning foundational concepts through advanced details.
Table 1: Proof Terminology & Logical Framework
The vocabulary of proof is the first thing to master: these terms define what you are trying to establish, what you may assume, and how the result relates to the broader mathematical landscape. Confusing a lemma with a theorem, or a conjecture with a proved result, leads to imprecise communication and reasoning.
| Term | Example | Description |
|---|---|---|
Axiom (postulate) | Euclid: "The whole is greater than the part" | • Self-evident foundational truth assumed without proof • everything in a formal system is derived from axioms |
Pythagorean theorem: a^2 + b^2 = c^2 | • Major proved mathematical result that can stand independently • the primary goal of a proof | |
Euclid's lemma: if p \mid ab and p is prime, then p \mid a or p \mid b | • Auxiliary result proved specifically to support proving a larger theorem • important on its own only incidentally | |
From \sqrt{2} irrational: \sqrt{2} is not expressible as p/q with p,q \in \mathbb{Z} | Easy, direct consequence of a theorem requiring little or no additional proof | |
Goldbach's conjecture: every even integer > 2 is the sum of two primes | • Unproven statement believed to be true • becomes a theorem once proved, or a disproven claim once a counterexample is found | |
"If n is odd, then n^2 is odd" | • A minor theorem of intermediate importance • stated and proved formally, but less significant than a theorem |