Mathematical language, notation and argumentAQA A-Level Maths: Revision notes

A constant is a fixed value, such as $5$ or $\pi$. A variable is a letter that can take different values, such as $x$. A term is one part of an expression, joined to others by $+$ or $-$, and the coefficient is the number multiplying the variable in a term. In $3x^2-5x+7$ the terms are $3x^2$, $-5x$ and $7$, the coefficient of $x$ is $-5$ and $7$ is the constant term. An expression has no equals sign, such as $x^2+2x$. An equation states two expressions are equal and is true only for some values, such as $x^2+2x=3$. An identity is true for every value of the variable and is written with $\equiv$, such as $(x+1)^2\equiv x^2+2x+1$. An index (power) shows repeated multiplication, as in $x^3$, where $3$ is the index. A function assigns exactly one output to each input.

Precise arguments use connecting symbols and words.

• $\Rightarrow$ means implies: $x=3\Rightarrow x^2=9$ (true), but the reverse fails because $x$ could be $-3$.
• $\Leftarrow$ means "is implied by", and $\Leftrightarrow$ means if and only if: both directions hold, as in $2x+1=7\Leftrightarrow x=3$.
• $\therefore$ means "therefore", used when concluding a line of working.
• $\equiv$ is used for identities. Words such as "so", "hence", "because" and "since" must connect lines logically. An argument should move from given assumptions through justified steps to a conclusion, and diagrams or sketched graphs are used where they help to justify a step. A statement such as $x^2=4\Rightarrow x=2$ is false as a deduction, because $x=-2$ also satisfies the left-hand side.

A set is a collection of elements written in braces, $\{1,2,3\}$. Set-builder notation $\{x\in\mathbb{R}:x>2\}$ means "the set of real $x$ such that $x>2$".

• $x\in A$: $x$ is an element of $A$. $x\notin A$: it is not.
• $A\cup B$ (union): in $A$ or $B$ or both. $A\cap B$ (intersection): in both.
• $A'$ (complement): everything in the universal set $\xi$ that is not in $A$. $\emptyset$ is the empty set. $A\subset B$: $A$ is a subset of $B$.
• Number sets: $\mathbb{N}$ (natural), $\mathbb{Z}$ (integers), $\mathbb{Q}$ (rational), $\mathbb{R}$ (real). Inequalities describe sets: with $A=\{x:-3<x<3\}$ and $B=\{x:x\ge1\}$, $A\cap B=\{x:1\le x<3\}$ and $A\cup B=\{x:x>-3\}$. In probability, $A\cap B$ is "$A$ and $B$" and $A\cup B$ is "$A$ or $B$", with $P(A\cup B)=P(A)+P(B)-P(A\cap B).$ Also $P(A')=1-P(A)$ and $(A\cup B)'=A'\cap B'$.

A function is a rule that gives exactly one output for each input. We write $f(x)=\ldots$ or $f:x\mapsto\ldots$. The domain is the set of allowed inputs and the range is the set of outputs that result. A relation where one input gives two outputs (such as $y=\pm\sqrt x$) is not a function. The domain may be restricted by the context or by the problem: $\sqrt{x-1}$ needs $x\ge1$ and $\frac{1}{x-2}$ needs $x\ne2$. Worked example: $f(x)=x^2-4x+7$ for $x\ge2$. Completing the square, $f(x)=(x-2)^2+3$. The minimum is at $x=2$, where $f=3$, and $f$ increases for $x\ge2$, so the range is $f(x)\ge3$. If the domain were $x\ge4$ the range would be $f(x)\ge7$, because $f(4)=7$. Changing the domain changes the range.

A good argument is valid at every line. When critiquing a proof or a solution, look for:

• steps that are only $\Rightarrow$ but are written as $\Leftrightarrow$, such as squaring both sides, which can add false roots ($\sqrt{x+2}=x$ gives $x=2$ and a false root $x=-1$);
• dividing by an expression that could be zero, which loses solutions: $x^2=3x$ gives $x=0$ or $x=3$, not just $x=3$;
• a claim "proved" from a few examples: examples support a statement but do not prove it, although one counter-example disproves it;
• answers that ignore the domain or context, such as a negative length;
• unjustified use of a diagram or graph. When you check a method or formula, ask what assumptions it needs and whether each step follows from the one before.