School of Mathematics:Introduction to Proofs:Lecture 1

Introduction

Exercise: Before reading any further, take a few minutes to try and define a proof for yourself.

A Proof is a sequence of logical deductions based on a set of assumptions, the axioms of the logic system, and previously proven statements of the system, ending at a desired conclusions. A proof is always associated with a Theorem, Lemma, or Corollary.

Less formally, given a statement, usually giving one or more assumptions and arriving at one or more conclusions, we prove that statement by starting with the givens and using what we know to eventually end up at the desired conclusions.

A theorem and its proof in mathematics must be distinguished from a theory (or law or etc.) in the sciences, in that a theorem in mathematics, once proven, is a fact (in mathematics) whereas a theory in the sciences is simply the best known explanation to date of a given pattern. In logical terms, a proof is deductive reasoning (meaning, if constructed properly, cannot be wrong) whereas a theory in the sciences is inductive reasoning (meaning it's probably correct, but may still be wrong).

Example

Theorem: Every even positive integer greater than 2 is not prime.

Proof: Suppose that we are given a positive integer $X$ greater than 2.

An even number is defined as a number which is an integer multiple of 2.

Therefore, that integer must be divisible by 2.

Since a prime number is a number which has no proper divisors, $X$ cannot be prime.

$\square$