Edexcel A-Level Mathematics

## What a proof is A **proof** starts from assumptions that are given or already established and moves through a series of logical steps to a conclusion, each step following from the ones before. A calculation that works for a few numbers is evidence, not a proof, because the claim is usually about every number in some set. A proof either covers the whole set at once, as algebra does, or covers every case one by one. Some notation makes the logic compact. The symbol $\Rightarrow$ means "implies", so $x = 3 \Rightarrow x^2 = 9$, but the reverse fails, since $x^2 = 9$ also allows $x = -3$. The symbol $\Leftrightarrow$ means "if and only if" and is used when each statement implies the other. An identity, written with $\equiv$, is true for every value of the variable, unlike an equation, which is true only for some. The letters $\mathbb{N}$, $\mathbb{Z}$, $\mathbb{Q}$ and $\mathbb{R}$ stand for the natural numbers, the integers, the rational numbers and the real numbers. ## Proof by deduction In **proof by deduction** you start from known facts and algebraic definitions and reason directly to the result. It is the method used throughout the subject, whenever a formula is derived. The usual first step is to write the objects in a general form: an even number is $2k$ and an odd number is $2k + 1$ for an integer $k$; consecutive integers are $n$ and $n + 1$; a multiple of 5 is $5k$. To prove that the sum of the squares of two consecutive integers is always odd, let the integers be…