Set Theory Explained — Meaning, Cardinality, Types & Complements (JAMB/WAEC Maths)

A set is a well-defined collection of distinct objects, and set theory provides the language and rules used to describe, classify, and compare these collections.

This topic is foundational in JAMB and WAEC Mathematics, underpinning later topics like Venn diagrams, since it introduces the basic vocabulary — cardinality, types of sets, and complements — used throughout set-based questions. This lesson covers all of these building blocks with clear definitions and examples.

Quick takeaways

• A set is a well-defined collection of distinct objects, called elements or members, usually written inside curly brackets.
• Cardinality, written n(A), refers to the number of elements in a set.
• Sets can be classified into types including finite, infinite, empty (null), universal, and equal sets.
• A subset is a set whose every element also belongs to another set; if two sets have no elements in common, they are called disjoint sets.
• The complement of a set includes every element in the universal set that is NOT in that specific set.

What is a set?

A set is a well-defined collection of distinct objects, called elements or members, typically written inside curly brackets. For example, the set of vowels can be written as A = {a, e, i, o, u}. “Well-defined” means it must always be clear whether a given object belongs to the set or not — this is what distinguishes a mathematical set from a vague or ambiguous grouping.

What is cardinality?

Cardinality refers to the number of elements contained within a set, written as n(A) for a set named A. For example, if A = {a, e, i, o, u}, then n(A) = 5, since the set contains 5 elements. Cardinality is frequently used in word problems, especially those solved using Venn diagrams, where you’re often asked to find n(A ∪ B) or n(A ∩ B) for two or more sets.

What are the main types of sets?

Several types of sets are commonly tested:

1. Finite set — contains a limited, countable number of elements (e.g., {1, 2, 3})
2. Infinite set — contains an unlimited number of elements (e.g., the set of all natural numbers)
3. Empty (null) set — contains no elements at all, written as { } or ∅
4. Universal set — the overall set containing all elements relevant to a particular discussion, usually denoted U
5. Equal sets — two sets containing exactly the same elements, regardless of order
6. Equivalent sets — two sets with the same number of elements (same cardinality), even if the elements themselves are different

What relationships can exist between sets?

Beyond classifying individual sets, set theory also describes how sets relate to one another:

• Subset — a set A is a subset of set B (written A ⊆ B) if every element of A is also an element of B
• Proper subset — a subset that doesn’t include every element of the larger set (i.e., A is a subset of B, but A ≠ B)
• Disjoint sets — two sets that share no elements in common at all

What is the complement of a set?

The complement of a set A, written A′ (or Aᶜ), includes every element in the universal set that is NOT a member of A. For example, if the universal set U = {1, 2, 3, 4, 5} and A = {1, 2}, then A′ = {3, 4, 5}. Understanding complements is essential for solving many Venn diagram problems, since questions frequently ask for the number of elements outside a specific set or combination of sets.

Common mistakes students make with set theory

• Confusing a subset (A ⊆ B) with set equality (A = B), when a subset only requires A’s elements to be contained within B, not identical to it
• Mixing up equal sets (identical elements) with equivalent sets (same number of elements, but not necessarily identical)
• Forgetting that the empty set is still considered a valid, finite set, with cardinality 0
• Calculating a complement incorrectly by forgetting to reference the correct universal set

Frequently asked questions

Q: What is cardinality in set theory? Cardinality, written n(A), refers to the number of elements contained within a set.

Q: What is the difference between equal sets and equivalent sets? Equal sets contain exactly the same elements, while equivalent sets simply have the same number of elements (the same cardinality), even if the actual elements differ.

Q: What is the complement of a set? The complement of a set includes every element in the universal set that is not a member of that specific set.

Q: What does it mean for two sets to be disjoint? Two sets are disjoint if they share no elements in common at all.

Ready to apply this to visual problems? Read our Venn Diagrams lesson →

This is one topic from our full JAMB & WAEC Maths course — see everything covered →