Set language is the notation used to describe collections of objects, test membership, and combine sets using union, intersection and complement. It is also the foundation for probability in Sec 4. Events are sets, and P(A ∪ B) only makes sense once union and intersection do.
In Paper 1, set questions are short and worth easy marks, but examiners penalise wrong notation heavily. Writing ∈ where ⊂ is needed, or {∅} where ∅ is meant, loses marks even when the idea is right. The symbols must be exact from the start.
What these notes cover
- •What is a set?
- •Membership: ∈ and ∉
- •Counting elements: n(A)
- •The empty set ∅
- •The universal set ξ
- •Listing form and set-builder form
- •Subsets and proper subsets
- •The complement A′
- •Intersection and union
- •Venn diagrams and shaded regions
- •Disjoint sets
- •Common mistakes and exam technique
Key definitions
- Set
- A well-defined collection of distinct objects, called its elements (or members). Well-defined means that for any object we can decide without ambiguity whether it belongs to the set. Order does not matter, and repeats are not counted.
- Empty set (∅)
- The set with no elements, written ∅ or {}, so n(∅) = 0. It is not the same as {0} or {∅}, each of which contains one element.
- Universal set (ξ)
- The set of all elements under consideration in a question. Every set in the question draws its elements from ξ, and ξ is drawn as the surrounding rectangle in a Venn diagram.
- Subset (A ⊆ B)
- A is a subset of B when every element of A is also an element of B. ∅ ⊆ A and A ⊆ A for every set A.
- Disjoint sets
- Two sets with no elements in common, so A ∩ B = ∅. On a Venn diagram they are drawn as two separate, non-overlapping circles.
Key formulas
Proper subset
Every element of A is in B, and B contains at least one extra element that A does not have. A set is never a proper subset of itself.
Complement
Everything in the universal set ξ that is not in A. A′ is meaningless unless ξ is stated.
Counting with the complement
Every element of ξ is in exactly one of A or A′.
Intersection
The elements in both A and B at the same time. This is the overlap of the two circles in a Venn diagram.
Union
“Or” is inclusive. Each element is listed once only, normally in ascending order.
Number of subsets
For a set with n elements. The count of subsets includes both ∅ and the set itself.
Worked example
It is given that ξ = {x : x is an integer, 1 ≤ x ≤ 12}, A = {x : x is a factor of 12} and B = {x : x is a multiple of 3}. List the elements of A and B, then find A ∩ B, A ∪ B and n(A ∪ B).
- 1List A by pairing the factors so that none are missed: 12 = 1 × 12 = 2 × 6 = 3 × 4, so A = {1, 2, 3, 4, 6, 12}.
- 2List B, stopping at 12 because ξ only runs up to 12: B = {3, 6, 9, 12}.
- 3Compare the two lists element by element; the common elements are 3, 6 and 12, so A ∩ B = {3, 6, 12}.
- 4Merge the two lists, writing each element once only and in ascending order: A ∪ B = {1, 2, 3, 4, 6, 9, 12}.
- 5Count the distinct elements of the union: n(A ∪ B) = 7.
Watch out: n(A ∪ B) is 7, not n(A) + n(B) = 6 + 4 = 10. The three shared elements must not be counted twice.
Common mistakes
Confusing ‘element of’ with ‘subset of’
Wrong: Writing 2 ⊆ A or {2} ∈ A, where A = {2, 4, 6}.
Right: Write 2 ∈ A and {2} ⊆ A. The symbol ∈ links an element to a set, and ⊆ links a set to a set. If there are no curly braces on the left, use ∈ (or ∉).
Thinking the empty set is {0}
Wrong: Writing A ∩ B = {0} or A ∩ B = 0 when two sets share no elements.
Right: Write A ∩ B = ∅. The set {0} contains one element, the number 0, so n({0}) = 1, whereas n(∅) = 0. A set also cannot equal a number.
Losing the elements outside both circles
Wrong: Leaving out the elements of ξ that belong to neither A nor B when completing a Venn diagram.
Right: Write them inside the rectangle but outside both circles. With ξ = {1, 2, 3, …, 10}, A = {2, 4, 6, 8, 10} and B = {3, 6, 9}, the elements 1, 5 and 7 go there. Omitting them makes later counts such as n(A′) wrong.
Questions students ask
What is the difference between a subset and a proper subset?
A ⊆ B means every element of A is also in B, and this allows A = B. A ⊂ B means A ⊆ B and A ≠ B, so B contains at least one extra element. Every set is a subset of itself, but never a proper subset of itself.
How many subsets does a set with n elements have?
A set with n elements has 2ⁿ subsets, counting both ∅ and the set itself, so it has 2ⁿ − 1 proper subsets. For example, {a, b, c} has 2³ = 8 subsets and 7 proper subsets. List them by size so that none are missed, especially ∅.
Why do you need the universal set to find a complement?
The complement A′ is everything in ξ that is not in A. Without ξ, “not in A” could include anything. For example, with ξ = {1, 2, 3, …, 10} and A = {2, 4, 6, 8, 10}, A′ = {1, 3, 5, 7, 9}.
How do you fill in a two-set Venn diagram?
Work from the inside out. Write the elements of A ∩ B in the overlap first, then put the remaining elements of A and of B in their own crescents. Any element of ξ still unused goes inside the rectangle but outside both circles. Then check that the four regions add up to n(ξ).
In the full PDF
- •Labelled Venn diagrams of the four standard shaded regions, including A′ and A ∩ B′
- •Three linked worked examples on one universal set, covering listing, Venn diagrams, describing shaded regions and shading (A ∪ B)′
- •Further worked examples: listing all subsets of {a, b, c}, finding a complement, and reading notation as true or false
- •The remaining classic errors, including shading the wrong region for a complement
- •Five exam-technique habits that earn the marks
- •A one-line symbol reference table and an end-of-topic summary