AQA A-Level Mathematics

## Notation and language A mathematical argument is built from precise statements joined by connecting words, so the symbols need to be used exactly. The symbol $\Rightarrow$ means "implies": $x = 3 \Rightarrow x^2 = 9$ is true, but the reverse is not, because $x$ could be $-3$. The symbol $\Leftrightarrow$ means "implies and is implied by", so that $2x = 6 \Leftrightarrow x = 3$. The symbol $\equiv$ marks an **identity**, which is true for every value of the variable, while $=$ in an **equation** is true only for particular values. The symbol $\therefore$ means "therefore". Set notation is used for solutions, inequalities and probability. | Symbol | Meaning | |---|---| | $x \in A$ | $x$ is an element of the set $A$ | | $A \subset B$ | $A$ is a subset of $B$ | | $A \cup B$ | union: in $A$ or in $B$ or in both | | $A \cap B$ | intersection: in both $A$ and $B$ | | $A'$ | the complement of $A$: everything in the universal set not in $A$ | | $\varnothing$ | the empty set | | $\{x : x > 2\}$ | the set of all $x$ such that $x > 2$ | | $\mathbb{N}$, $\mathbb{Z}$, $\mathbb{Q}$, $\mathbb{R}$ | natural numbers, integers, rational numbers, real numbers | The solution of $x^2 > 4$ can be written $x < -2$ or $x > 2$, or as the set $\{x : x < -2\} \cup \{x : x > 2\}$. The solution of $2 < x < 5$ is the intersection $\{x : x > 2\} \cap \{x : x < 5\}$. "And" corresponds to an intersection and "or" to a union. A **rational number** can be written as $\frac{p}{q}$ with $p$ and $q$ integers…