CCEA A-Level Mathematics

## Symbols and connecting language Mathematical arguments are written with a small set of symbols, and each one has a precise meaning that a sentence of working has to respect. | Symbol | Meaning | |---|---| | $=$ | is equal to, for particular values | | $\equiv$ | is identically equal to, true for every value of the variable | | $\neq$ | is not equal to | | $\therefore$ | therefore | | $\because$ | because | | $\Rightarrow$ | implies, so if the left side is true the right side is true | | $\Leftarrow$ | is implied by | | $\Leftrightarrow$ | implies and is implied by, so the two statements are equivalent | The difference between $=$ and $\equiv$ matters. The statement $x^2 - 1 = 0$ is true only for $x = 1$ and $x = -1$, whereas $x^2 - 1 \equiv (x - 1)(x + 1)$ is true whatever $x$ is. The arrow $\Rightarrow$ goes in one direction only: $x = 3 \Rightarrow x^2 = 9$ is true, but $x^2 = 9 \Rightarrow x = 3$ is false because $x$ could be $-3$. Since $x^2 = 9 \Leftrightarrow x = 3 \text{ or } x = -3$, the equivalence sign can be used there. A condition is **necessary** for a statement if the statement cannot be true without it, and **sufficient** if it guarantees the statement. Being a multiple of $4$ is sufficient for a whole number to be even but not necessary, because $6$ is even and is not a multiple of $4$. Being even is necessary for being a multiple of $4$ but not sufficient, since $6$ is even and is not a multiple of $4$. When a condition is both, the two statements are…