What does \Leftrightarrow mean?

If and only if: the statements imply each other.

What does \therefore mean?

Therefore: introduces a conclusion.

Define the domain of a function.

The set of allowed input values.

Define the range of a function.

The set of output values the function produces.

What does A\cap B mean?

The set of elements in both A and B (intersection).

What does A\cup B mean?

The set of elements in A or B or both (union).

The complement: elements of the universal set not in A.

Set equal to A'\cap B'?

(A\cup B)', the complement of the union.

Which sets are written \mathbb{N},\mathbb{Z},\mathbb{Q},\mathbb{R}?

Natural numbers, integers, rational numbers and real numbers.

Why can squaring both sides of an equation produce false roots?

The step is only \Rightarrow: 1^2=(-1)^2, so a value that satisfies the squared equation may not satisfy the original.

Why is x^2=3x\Rightarrow x=3 wrong?

Dividing by x loses the solution x=0. Factorise instead: x(x-3)=0.

Mathematical language, notation and argumentAQA A-Level Maths: Flashcards

Card 1 of 130 of 13 known

Question

What is the difference between an equation and an identity?

Tap or press Space to reveal

Tap card or press Space to flip

See all 13 cards

What is the difference between an equation and an identity?: An equation is true only for particular values. An identity ( $\equiv$ ) is true for every value.
What does $\Leftrightarrow$ mean?: If and only if: the statements imply each other.
What does $\therefore$ mean?: Therefore: introduces a conclusion.
Define the domain of a function.: The set of allowed input values.
Define the range of a function.: The set of output values the function produces.
What does $A\cap B$ mean?: The set of elements in both $A$ and $B$ (intersection).
What does $A\cup B$ mean?: The set of elements in $A$ or $B$ or both (union).
What does $A'$ mean?: The complement: elements of the universal set not in $A$ .
Formula for $P(A\cup B)$ ?: $P(A)+P(B)-P(A\cap B)$
Set equal to $A'\cap B'$ ?: $(A\cup B)'$ , the complement of the union.
Which sets are written $\mathbb{N},\mathbb{Z},\mathbb{Q},\mathbb{R}$ ?: Natural numbers, integers, rational numbers and real numbers.
Why can squaring both sides of an equation produce false roots?: The step is only $\Rightarrow$ : $1^2=(-1)^2$ , so a value that satisfies the squared equation may not satisfy the original.
Why is $x^2=3x\Rightarrow x=3$ wrong?: Dividing by $x$ loses the solution $x=0$ . Factorise instead: $x(x-3)=0$ .

Exam questions on Mathematical language, notation and argument

Let $A=\{x\in\mathbb{R}:x^2<9\}$ and $B=\{x\in\mathbb{R}:x\ge1\}$ . The universal set is $\mathbb{R}$ and $A'$ denotes the complement of $A$ .
Write down $A'\cap B$ as an inequality, showing how you obtain it.2 marks
The function $f$ is defined by $f(x)=x^2-4x+7$ for $x\in\mathbb{R}$ , $x\ge2$ .
The domain of $f$ is changed to $x\ge4$ . Find the range of $f$ for this new domain, justifying your answer.2 marks
A student solves $\sqrt{x+2}=x$ and writes: Line 1: $x+2=x^2$ . Line 2: $x^2-x-2=0$ . Line 3: $(x-2)(x+1)=0$ . Line 4: so $x=2$ or $x=-1$ .
Show that $x=-1$ is not a solution of the original equation, and explain how it arose.3 marks

See the full worksheet

Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).