A set is a collection of objects, numbers or ideas that belong together. Each object in a set is called a member or an element. For example, the colours of the flag of Trinidad and Tobago form the set {red, white, black}. Sets help us sort and classify things, and they are the starting point for Venn diagrams, probability and logic. This lesson covers naming and describing sets, the main types of sets, subsets, and the ideas of union, intersection and complement shown on Venn diagrams.

Naming and describing a set

• Name a set with a capital letter and put its members inside curly brackets { }, separated by commas: V = {a, e, i, o, u}.
• A set must be well defined: you must be able to say for certain whether something belongs. “Even numbers” is well defined; “nice songs” is not.
• You can define a set in two ways: by listing the members, V = {a, e, i, o, u}, or by describing them in words, V = {vowels in the English alphabet}.
• The symbol ∈ means “is a member of” and ∉ means “is not a member of”: e ∈ V, but b ∉ V.
• n(A) means the number of members in set A. For the vowels, n(V) = 5.
• Each member is listed only once, and the order does not matter: {1, 2, 3} is the same set as {3, 1, 2}.

Types of sets

Type	Meaning	Example
Empty (null) set	Has no members. Written { } or ∅.	{months with 32 days} = { }
Finite set	You could count all its members.	{pupils in your class}
Infinite set	The members go on forever. Use “…” in a list.	{multiples of 5} = {5, 10, 15, …}
Equal sets	Exactly the same members (A = B).	{letters in LISTEN} = {letters in SILENT}
Equivalent sets	The same number of members, n(A) = n(B), even if the members differ.	{1, 2, 3} and {a, b, c}

Subsets

If every member of set A is also in set B, then A is a subset of B, written A ⊂ B (some books write A ⊆ B). For example, {2, 4} ⊂ {1, 2, 3, 4, 5}. Two useful facts: the empty set is a subset of every set, and every set is a subset of itself. When we classify the members of a set, we divide it into subsets: {1, 2, 3, 4, 5, 6} can be split into the subsets {odd numbers} = {1, 3, 5} and {even numbers} = {2, 4, 6}.

Universal set, complement, union and intersection

• The universal set, U, contains every member being talked about in a question – for example U = {whole numbers from 1 to 12}.
• The complement of A, written A′, is everything in U that is not in A.
• The union A ∪ B is everything in A or B or both (join the sets together, without repeats).
• The intersection A ∩ B is everything in both A and B (the overlap).
• Sets that share at least one member are intersecting sets. Sets with no members in common are disjoint: A ∩ B = { }.

Venn diagrams

A Venn diagram shows sets as circles inside a rectangle that stands for U. Overlapping circles show intersecting sets; separate circles show disjoint sets; a circle drawn completely inside another shows a subset. Members outside every circle are still in U – do not forget to write them in the rectangle.

Worked examples

Example 1: Listing a set

Question: List the members of A = {odd numbers between 2 and 12} and find n(A).

1. “Between 2 and 12” means we do not include 2 or 12.
2. The odd numbers in that range are 3, 5, 7, 9, 11.

Answer: A = {3, 5, 7, 9, 11}, n(A) = 5.

Example 2: Describing a set in words

Question: Describe the set {2, 4, 6, 8} in words.

1. Look for what the members have in common: they are all even, start at 2 and stop at 8.

Answer: {even numbers from 2 to 8} (or {the first four even numbers}).

Example 3: Union, intersection and complement

Question: U = {whole numbers from 1 to 12}, A = {factors of 12} and B = {even numbers}. List A, B, A ∩ B, A ∪ B, A′ and (A ∪ B)′, and draw the Venn diagram.

1. A = {1, 2, 3, 4, 6, 12} and B = {2, 4, 6, 8, 10, 12}.
2. A ∩ B (in both) = {2, 4, 6, 12}. These go in the overlap.
3. A only = {1, 3}; B only = {8, 10}.
4. A ∪ B = {1, 2, 3, 4, 6, 8, 10, 12}.
5. A′ (in U but not in A) = {5, 7, 8, 9, 10, 11}.
6. (A ∪ B)′ (outside both circles) = {5, 7, 9, 11}.
7. Check: n(A) + n(B) − n(A ∩ B) = 6 + 6 − 4 = 8 = n(A ∪ B) ✔

Answer: Venn diagram: 1 and 3 in A only; 2, 4, 6, 12 in the overlap; 8 and 10 in B only; 5, 7, 9, 11 outside the circles. (Choose “Factors of 20”, “Even numbers” and other rules in the builder above to practise more of these.)

Example 4: Equal or equivalent?

Question: Are {letters in LISTEN} and {letters in SILENT} equal sets? Are {1, 2, 3} and {a, b, c} equal?

1. LISTEN gives {L, I, S, T, E, N} and SILENT gives {S, I, L, E, N, T} – the same six letters, so the sets are equal.
2. {1, 2, 3} and {a, b, c} have different members, but both have 3 members.

Answer: The first pair are equal sets. The second pair are equivalent but not equal.

Example 5: Listing all subsets

Question: List every subset of {p, q, r}.

1. No members: { }.
2. One member: {p}, {q}, {r}.
3. Two members: {p, q}, {p, r}, {q, r}.
4. Three members: {p, q, r}.

Answer: 8 subsets. (A set with n members has 2ⁿ subsets: 2³ = 8.)

Common mistakes

• Repeating members. The letters in “BANANA” form the set {B, A, N}, not {B, A, N, A, N, A}.
• Mixing up ∪ and ∩. Think “∪ for union = everything joined”; “∩ looks like an n for intersection = only the overlap”.
• Forgetting members outside the circles. Every member of U must appear somewhere in the rectangle.
• Confusing ∅ with {0}. ∅ has no members; {0} has one member, the number 0.
• Using ∈ for sets. 3 ∈ {1, 3, 5} (a member), but {3} ⊂ {1, 3, 5} (a subset).

Practice questions

Try these yourself first, then open each answer to check your working.

Quick check

Next: Tally Charts, Frequency Tables, Pictographs and Block Graphs

Curriculum link: This post is relevant to the Trinidad and Tobago National Lower Secondary (NCSE) Mathematics Curriculum – Form 1, Term 1, Strand 1.2 Sets, Relations and Functions, Topic 1.2.1 Sets – Learning Outcomes 1.2.1.1 (classifying, describing and naming sets), 1.2.1.2 (defining sets by listing or describing), 1.2.1.3 (dividing a set into subsets), 1.2.1.4 (empty, equal, equivalent, finite and infinite sets), 1.2.1.5 (universal sets, complements, union, intersecting sets, subsets and disjoint sets) and 1.2.1.6 (Venn diagrams for two sets).

Attributions to media used in this post:

Venn-stainedglass-gonville-caius.jpg – Schutz (photo); stained glass designed by Maria McClafferty – CC BY-SA 2.5 (via Wikimedia Commons)

Updates

2026/10/25

– Put each lettered part of the questions and answers on its own line.

© 2026 Vedesh Kungebeharry. All rights reserved.