Advanced Mathematics

1.4Mathematical Proofs

Purpose of Section

Most theorems in mathematics take the form of a conditional P ⇒ Q or biconditional statement P ⇔ Q, where the biconditional can be verified by proving both P ⇒ Q and P ⇐ Q. In this section, we describe several different ways for proving P ⇒ Q including a direct proof, proof by contrapositive, and three variations of proof by contradiction, one of which is called proof by reductio ad absurdum.

1.4.1 Introduction

Proof is basic to mathematics; we do not know if a proposition is true or false until we have proved or disproved it, which raises the obvious question, what is a mathematical proof? The precise definition of mathematical proof varies from mathematician to mathematician. The famed mathematician GianCarlo Rota once remarked: “Everybody knows what a mathematical proof is, it's a series of steps which leads to the desired conclusion.” A more rigid viewpoint of proof might be manipulating definitions and accepted rules of logic in a valid way, going from the assumption to the conclusion. But regardless of its definition, the history of what constitutes a mathematical proof has gone through several refinements over the years, each refinement attaining a higher level of “rigor” from its predecessors.¹ Some mathematical proofs proposed by such greats as Newton and Euler² do not hold up to today's scrutiny.

As an example of a mathematical proof, consider the proposition:

If a, b, c are real numbers, a ≠ 0 that satisfy ax