Subsets and set notation revision notes

Edexcel International GCSE Maths (4MA1), chapter 18: Sets, specification point 1.5a. Free lesson.

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Set notation describes which elements belong to a set, while subset symbols compare whole sets. Listing the elements first makes intersections, unions and complements easier to work out.

What a subset means

A set A is a subset of B if every element in A is also in B. The symbol ⊆ allows A to have exactly the same elements as B. The symbol ⊂ means A is inside B but is smaller than B. The empty set, ∅, is a subset of every set. A ⊆ B allows A = B. A ⊂ B means A is strictly smaller than B. ∅ ⊆ A for every set A.

How to read set-builder notation

In set-builder notation, the colon means “such that”. What follows the colon is a rule, or filter, that decides which elements are included. The universal set is the full collection of elements you are allowed to choose from. List the elements that pass the filter before doing anything else with the set. For example, {x : x is a factor of 12} means the set of factors of 12. If the universal set is the whole numbers from 1 to 12, then A = {1, 2, 3, 4, 6, 12} for factors of 12, and B = {2, 4, 6, 8, 10, 12} for even numbers.

How to use set symbols

The symbols show whether an element belongs to a set, or how sets relate to each other. ∩ means elements in both sets. ∪ means elements in either set, including those in both, and each element is listed only once. The complement B′ means everything in the universal set that is not in B. A ∩ B = {2, 4, 6, 12}. A ∪ B = {1, 2, 3, 4, 6, 8, 10, 12}. B′ = {1, 3, 5, 7, 9, 11}, using the universal set {1, 2, …, 12}.

How to choose between ∈ and ⊆

Use ∈ for one element on its own. Use ⊆ for a whole set. So 3 ∈ A says that the number 3 belongs to A, while {3} ⊆ A says that the set containing 3 is a subset of A. These statements use different symbols because one is about a number and the other is about a set. 3 ∈ A. {3} ⊆ A.

How to find a complement

To find P′, use the universal set and leave out every element that belongs to P. The universal set matters because it sets the limits of the answer.