OCR A-Level Mathematics A

## What a proof is A **proof** starts from assumptions and facts already established and moves through logical steps to a conclusion, with nothing left to trust. A single worked example never proves a general statement, because the next case might behave differently. A proof has to cover every case the statement claims. The vocabulary matters. An **integer** is a whole number (positive, negative or zero), a **rational** number can be written as $\dfrac{p}{q}$ with $p$ and $q$ integers and $q \ne 0$, and a **real** number is any point on the number line. A real number that is not rational is **irrational**. The symbol $\mathbb{Z}$ stands for the integers, $\mathbb{Q}$ for the rationals and $\mathbb{R}$ for the reals. ## Logical connectives Three symbols connect statements. $P \Rightarrow Q$ means "if $P$ then $Q$": whenever $P$ is true, $Q$ is true as well. $P \Leftarrow Q$ is the same statement the other way round. $P \Leftrightarrow Q$ means both hold, and is read "if and only if" (often shortened to "iff"). It says the two statements are equivalent. The two directions are separate claims and one does not give the other. If $x = 3$ then $x^2 = 9$, so $x = 3 \Rightarrow x^2 = 9$. The reverse fails, because $x^2 = 9$ is also true for $x = -3$. So $x^2 = 9$ does not imply $x = 3$. When you solve an equation and each line follows from the last in both directions, you can write $\Leftrightarrow$; squaring both sides is a typical step where only $\Rightarrow$ is safe and false…