Sets (CSEC Mathematics)

A set is a well-defined collection of distinct objects, called elements or members. "Well-defined" means it is always clear whether an object belongs to the set or not. Sets are one of the most useful ideas in CSEC Mathematics because they give you a precise language for grouping things and for solving "how many" problems with Venn diagrams.

Set notation

A set is written with curly brackets: A = {2, 4, 6, 8}.
∈ means "is an element of": 4 ∈ A. ∉ means "is not an element of": 5 ∉ A.
n(A) is the cardinality — the number of elements in A. Here n(A) = 4.
Order and repetition do not matter: {2, 4, 6} is the same set as {6, 4, 2, 2}.
Sets can be described by listing ( {1, 2, 3} ) or by a rule/set-builder form, e.g. {x : x is an even number, x < 10}.

Types of sets

Finite set — has a countable number of elements, e.g. {days of the week}.
Infinite set — has unlimited elements, e.g. {whole numbers}.
Empty (null) set — has no elements, written ∅ or { }. Note: {0} is not empty — it contains the element 0.
Universal set (U) — the set of all elements under consideration in a problem.
Equal sets — contain exactly the same elements: {1, 2, 3} = {3, 2, 1}.
Subset (⊆) — every element of B is also in A means B ⊆ A. The empty set is a subset of every set, and every set is a subset of itself.
Proper subset (⊂) — B ⊂ A means B is a subset of A but B ≠ A.

Set operations

For these, picture the regions of a Venn diagram.

Union (A ∪ B) — all elements in A or B (or both). "Everything in either circle."
Intersection (A ∩ B) — elements in A and B (the overlap). If A ∩ B = ∅ the sets are disjoint.
Complement (A′) — all elements in the universal set U that are not in A.
Difference (A − B or A \ B) — elements in A that are not in B.

Worked example. U = {1,2,3,4,5,6,7,8}, A = {1,2,3,4}, B = {3,4,5,6}.

A ∪ B = {1,2,3,4,5,6}
A ∩ B = {3,4}
A′ = {5,6,7,8}
A − B = {1,2}

Venn diagrams

A Venn diagram shows the universal set as a rectangle and each set as a circle inside it. Overlapping circles share an intersection region. With two sets there are four regions: only A, only B, both (A ∩ B), and neither (outside both). With three sets there are eight regions — take care to fill the centre (A ∩ B ∩ C) first and work outwards.

Solving problems with Venn diagrams

This is the most heavily examined part of the topic. The reliable method:

Draw the diagram and label the universal set.
Start with the innermost overlap (the "both"/"all three" region) and write that number in first.
Work outwards, subtracting values you have already placed so each region shows only the people/items unique to it.
Use n(A ∪ B) = n(A) + n(B) − n(A ∩ B) to check, or to find a missing value.

Worked example (two sets). In a class of 30 students, 18 study French, 15 study Spanish, and 5 study both.

Both = 5 (place in the overlap).
French only = 18 − 5 = 13.
Spanish only = 15 − 5 = 10.
Neither = 30 − (13 + 5 + 10) = 2.

You can check: n(F ∪ S) = 18 + 15 − 5 = 28, and 30 − 28 = 2 study neither. ✓

Tip for "x" problems. When an unknown number do both, let the overlap be x, write each region in terms of x, add them to the total, and solve the equation. For example, if 18 do French, 15 do Spanish, 2 do neither, in a class of 30: (18 − x) + x + (15 − x) + 2 = 30 → 35 − x = 30 → x = 5.

Common mistakes to avoid

Forgetting to subtract the overlap, so the regions double-count the "both" group.
Confusing ∪ (union, "or") with ∩ (intersection, "and").
Treating ∅ and {0} as the same — they are not.
Leaving out the "neither" region when totalling.

Exam tips

Read whether the question asks for "only A", "A and B", or "A or B" — the wording decides the region.
Always check your regions add up to the universal set total.
Show the Venn diagram even if not explicitly asked — method marks are awarded for it.