Set notation for A-level maths

Set notation in A-level maths is a way of writing a collection of numbers with curly brackets and a few symbols. The one you will use most is {x : x > 3}, read as "the set of all x such that x is greater than 3". A-level questions often ask for the solution of an inequality in set notation: x < −2 or x > 5 becomes {x : x < −2} ∪ {x : x > 5}.

Set notation symbols

These are the symbols from the set notation list in the AQA A-level Maths specification. Other boards use the same ones.

Some books write a vertical line instead of a colon, {x | x > 3}. It means the same thing. In the US this style is called set-builder notation.

Number sets

So {x ∈ ℝ : x > 3} is the set of real numbers greater than 3, and {x ∈ ℤ : x > 3} is {4, 5, 6, ...}. If no number set is given, assume x is real.

Writing inequalities in set notation

AQA's specification says solutions of inequalities can be given using "and" and "or" correctly, or in set notation. Set notation is the safest choice, as it can only be read one way. There are three shapes of answer:

The rule to remember: "or" becomes ∪ (union: either one) and "and" becomes ∩ (intersection: both at once).

Example 1: a linear inequality

Solve 5x − 3 > 2x + 9, giving your answer in set notation.

Answer: {x : x > 4}.

Example 2: a quadratic inequality with two pieces

Solve x² − 3x − 10 > 0. Factorise to find the critical values:

Here x is the input and y = x² − 3x − 10. Its roots are −2 and 5, and y is positive to the left of −2 and to the right of 5. So x < −2 or x > 5. In set notation: {x : x < −2} ∪ {x : x > 5}.

Example 3: a quadratic inequality with one piece

Solve x² − x − 12 ≤ 0. It factorises as (x − 4)(x + 3) ≤ 0. This parabola is negative or zero between its roots, so −3 ≤ x ≤ 4. In set notation: {x : −3 ≤ x ≤ 4}, which you can also write as {x : x ≥ −3} ∩ {x : x ≤ 4}.

You can check any of these in the Inequality Solver, which shows the solution on a number line.

Check your understanding.

Interval notation

Intervals of real numbers can also be written with brackets. A square bracket includes the end point and a round bracket leaves it out:

So {x : −3 ≤ x ≤ 4} is the interval [−3, 4]. Be careful: (a, b) also means a coordinate pair, so only use it for an interval when the meaning is clear. The interval notation guide covers this in more detail.

Set notation in probability

In A-level statistics, events are sets, and the same symbols describe them:

• A ∪ B: A or B or both happens.
• A ∩ B: A and B both happen.
• A′: A does not happen.
• ξ: the universal set, everything that can happen. Some books use ℰ or S.

From the Venn diagram: P(A) = 0.4 + 0.1 = 0.5, P(B) = 0.1 + 0.2 = 0.3, P(A ∩ B) = 0.1, and P(A ∪ B) = 0.7. These fit the addition rule:

The complement of A ∪ B is the region outside both circles: P((A ∪ B)′) = 1 − 0.7 = 0.3.

Domain and range in set notation

The domain of a function can be written as a set. For f(x) = √(x − 2), you can only take the square root of a number that is not negative, so the domain is {x ∈ ℝ : x ≥ 2}. Its range is {y ∈ ℝ : y ≥ 0}, which A-level papers usually write as f(x) ≥ 0.

Common mistakes

• Using ∩ for "or". Two separate pieces need ∪.
• Writing {−2, 5} for an inequality. That is the set of just two numbers, −2 and 5.
• Listing both pieces in one bracket, such as {x : x < −2, x > 5}. Use two sets joined by ∪.
• Changing ≤ to <. Keep the inequality signs exactly as they are in your solution.
• Leaving out the "x :". {x > 4} is not correct set notation; write {x : x > 4}.

Practice questions

1. Write x ≤ 7 in set notation.
2. Solve 2x + 5 < 3, giving your answer in set notation.
3. Solve x² − 4x − 21 ≥ 0, giving your answer in set notation.
4. List the elements of {x ∈ ℤ : −2 < x ≤ 3}.

Answers: 1) {x : x ≤ 7}. 2) 2x < −2, so {x : x < −1}. 3) (x − 7)(x + 3) ≥ 0, so {x : x ≤ −3} ∪ {x : x ≥ 7}. 4) {−1, 0, 1, 2, 3}.

Practise with a short timed paper

Inequalities in set notation come up in A-level Maths pure papers, and set notation for events in statistics. The daily A-level Maths: Algebra & Functions and A-level Maths: Statistics papers are short timed papers of 8 questions, with a new one every day.

Read next

Maths · AlgebraInterval notation: brackets, infinity and examplesInterval notation explained: when to use square or round brackets, how to write infinity and unions, and how to turn inequalities into intervals.