A complete guide to set notation, Venn diagrams, union, intersection, complement, subsets, and solving survey-style counting problems.
A set is simply a well-defined collection of distinct objects, called its elements or members. Set theory provides precise notation and a visual tool — the Venn diagram — for describing how different groups of objects relate, overlap, and combine. These ideas underpin probability, logic, and data organisation, and appear constantly in real-world contexts such as surveys, classifications, and database queries.
A set is written using curly braces, listing its elements: A = {2, 4, 6, 8}. Sets can also be described by a rule rather than a full list, e.g. A = {even numbers less than 10}.
| Symbol | Meaning |
|---|---|
| ∈ | "is an element of" (e.g. 4 ∈ A) |
| ∉ | "is not an element of" |
| n(A) | The number of elements in set A |
| ⊂ | "is a subset of" |
| ∅ or { } | The empty set (no elements) |
| ξ (or U) | The universal set (everything under consideration) |
The universal set, ξ, contains every element relevant to a given problem — all other sets are considered subsets of it. The empty set, written ∅ or { }, contains no elements at all.
Set A is a subset of set B (written A ⊂ B) if every element of A is also an element of B. Every set is a subset of the universal set, and the empty set is a subset of every set.
A = {1, 3, 5}, B = {1, 2, 3, 4, 5, 6}. Is A ⊂ B?
Every element of A (1, 3, 5) is also in B, so yes, A ⊂ B.
A Venn diagram represents sets as overlapping circles inside a rectangle (the universal set ξ). Elements are placed in the region corresponding to which sets they belong to.
The union of A and B, written A ∪ B, contains every element that is in A, B, or both. On a Venn diagram, this is the entire shaded area covered by both circles combined.
A = {1, 2, 3, 4}, B = {3, 4, 5, 6}. Find A ∪ B.
A ∪ B = {1, 2, 3, 4, 5, 6} (each element listed once, even though 3 and 4 appear in both).
The intersection of A and B, written A ∩ B, contains only the elements that are in both A and B. On a Venn diagram, this is just the overlapping region.
Using the same sets, A = {1, 2, 3, 4}, B = {3, 4, 5, 6}. Find A ∩ B.
A ∩ B = {3, 4} — the elements common to both sets.
The complement of A, written A′, contains every element of the universal set that is not in A. On a Venn diagram, this is everything outside circle A but still inside the rectangle.
ξ = {1, 2, 3, 4, 5, 6, 7, 8}, A = {2, 4, 6, 8}. Find A′.
A′ = {1, 3, 5, 7} — everything in ξ that is not in A.
A useful formula connects the sizes of two sets, their union, and their intersection: n(A ∪ B) = n(A) + n(B) − n(A ∩ B). The intersection is subtracted because its elements would otherwise be counted twice.
In a class, 18 students study French, 15 study Spanish, and 7 study both. Find the number studying French or Spanish.
n(F ∪ S) = 18 + 15 − 7 = 26 students.
The standard approach for a word problem is to fill in the intersection region first (since it is shared information), then work outward to fill in the parts of each set that are not shared, and finally the region outside both circles (but inside ξ).
In a survey of 40 people, 22 like tea, 25 like coffee, and 12 like both. How many like neither?
Number liking only tea = 22 − 12 = 10. Number liking only coffee = 25 − 12 = 13. Total liking at least one = 10 + 12 + 13 = 35. Number liking neither = 40 − 35 = 5 people.
Problems can also involve three overlapping sets, requiring a diagram with three circles and up to eight distinct regions (including the region outside all three sets). The same principle applies: fill in the most specific (most overlapping) region first, then work outward.
Common errors include: listing shared elements twice when writing a union; forgetting to subtract the intersection when using the n(A ∪ B) formula; confusing union (∪, "or") with intersection (∩, "and"); and filling a Venn diagram from the outside in rather than starting with the most specific overlapping region.
| Notation | Meaning |
|---|---|
| A ∪ B | Elements in A, B, or both |
| A ∩ B | Elements in both A and B |
| A′ | Elements not in A (within ξ) |
| n(A ∪ B) | n(A) + n(B) − n(A ∩ B) |
Every set problem reduces to correctly identifying which regions of a Venn diagram are involved, then applying the matching notation or counting formula.
This article is original EDUSAMBAM educational writing. It is designed as a broad sets and Venn diagrams resource covering the notation and problem-solving techniques needed for confident set theory work. Exact examination requirements can vary between examination boards and syllabuses, so students should also compare their work with the specification and past-paper requirements of their own board.
Recommended study approach: memorise the set notation table, practise shading union, intersection and complement regions on two-circle Venn diagrams, then work through survey-style word problems by filling the most overlapping region first.
20 questions covering set notation, subsets, union, intersection, complement, and Venn diagram word problems. Answer every question, then submit to see your score instantly.