Sets: Notation, Types of Sets, Subsets, Set Operations and Venn Diagram Problems

Why study sets?

Sets are the language of sorting and grouping. A search engine that finds pages containing "WAEC" and "physics" is using an intersection; a playlist that combines two albums is a union. Databases, computer programs and probability all rest on sets, and almost every WAEC Further Mathematics paper has a Venn diagram question.

What is a set?

A set is a well-defined collection of distinct objects, called its elements (or members).

Well-defined means everyone can agree whether an object belongs. "Vowels of the English alphabet" is a set: A = {a, e, i, o, u}. "Beautiful houses" or "brilliant students" is not a set, because beauty and brilliance are opinions.

• Sets are named with capital letters and their elements are listed inside curly brackets { }.
• Each element is listed once: {1, 2, 2, 3} is just {1, 2, 3}.
• Order does not matter: {3, 1, 2} = {1, 2, 3}.

Set notation

• ∈ means "is an element of": 2 ∈ {2, 4, 6}.
• ∉ means "is not an element of": 9 ∉ {2, 4, 6}.
• n(A) is the cardinal number of A, the number of elements: if A = {2, 4, 6} then n(A) = 3.
• ∅ or { } is the empty set, and n(∅) = 0.

A set can contain another set as an element: if A = {1, 2, {3, 4}}, then {3, 4} ∈ A and n(A) = 3.

Three ways to describe a set

1. Listing (roster) method: A = {2, 4, 6, 8}
2. Description: A is the set of even numbers less than 10
3. Set-builder notation: A = {x : x is an even number, x < 10}

Read the colon as "such that". Set-builder notation is the best way to describe large or infinite sets, for example {x : x ∈ ℝ, −2 ≤ x < 5}.

Types of sets

• Finite set: has a definite number of elements, e.g. {January, February, March}.
• Infinite set: never ends, e.g. the natural numbers {1, 2, 3, …}.
• Empty (null) set ∅: has no elements, e.g. months with 32 days.
• Singleton set: has exactly one element, e.g. {the smallest prime number} = {2}.
• Equal sets: contain exactly the same elements: {1, 2, 3} = {3, 2, 1}.
• Equivalent sets: have the same number of elements: {a, b, c} and {2, 4, 6} are equivalent but not equal.
• Universal set U (or ξ): contains every element under discussion, drawn as the rectangle in a Venn diagram.
• Disjoint sets: have no elements in common, so A ∩ B = ∅.

Subsets and power sets

A is a subset of B, written A ⊆ B, if every element of A is also in B. For example {2, 4} ⊆ {2, 4, 6, 8}.

A is a proper subset of B, written A ⊂ B, if A ⊆ B and A ≠ B (B has at least one extra element).

• The empty set is a subset of every set.
• Every set is a subset of itself, but not a proper subset of itself.

The power set P(A) is the set of all subsets of A. A set with n elements has:

• 2ⁿ subsets
• 2ⁿ − 1 proper subsets

Operations on sets

• Union A ∪ B: every element in A or B (or both). If A = {1, 2, 3} and B = {3, 4, 5}, then A ∪ B = {1, 2, 3, 4, 5}.
• Intersection A ∩ B: only the elements in both A and B. Here A ∩ B = {3}.
• Complement A′: the elements of U that are not in A.
• Difference A − B (= A ∩ B′): the elements in A but not in B. Here A − B = {1, 2}.

Laws of set algebra

• Commutative: A ∪ B = B ∪ A and A ∩ B = B ∩ A
• Associative: (A ∪ B) ∪ C = A ∪ (B ∪ C), and the same for ∩
• Distributive: A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) and A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
• De Morgan's laws: (A ∪ B)′ = A′ ∩ B′ and (A ∩ B)′ = A′ ∪ B′
• Identities: A ∪ ∅ = A, A ∩ ∅ = ∅, A ∪ A = A, A ∩ A = A, A ∪ A′ = U, A ∩ A′ = ∅

Venn diagrams

In a Venn diagram the rectangle is the universal set and each circle is a set. Fill in a Venn diagram from the inside out: start with the region common to all the sets, then subtract.

For two sets: n(A ∪ B) = n(A) + n(B) − n(A ∩ B)

For three sets: n(A ∪ B ∪ C) = n(A) + n(B) + n(C) − n(A ∩ B) − n(B ∩ C) − n(A ∩ C) + n(A ∩ B ∩ C)

When the number in a region is unknown, call it x, write every region in terms of x, then use the total.

Common mistakes

1. Adding n(A) and n(B) without subtracting the overlap, so the middle region is counted twice.
2. Filling the "Physics" circle with 25 straight away instead of starting from the centre.
3. Forgetting the region outside all the circles (inside U).
4. Mixing up ⊂ (proper subset) and ∈ (element of): 2 ∈ {1, 2} but {2} ⊂ {1, 2}.
5. Saying a set with 4 elements has 8 subsets: it has 2⁴ = 16.

Practice

1. List the elements of {x : x is a factor of 18}.
2. How many proper subsets has {p, q, r, s}?
3. U = {1, 2, …, 10}, A = {x : x is odd} and B = {x : x is prime}. Find A ∩ B and (A ∪ B)′.
4. If n(A) = 15, n(B) = 12 and n(A ∪ B) = 20, find n(A ∩ B).
5. In a group of 60 people, 35 read newspaper X, 30 read newspaper Y and 10 read neither. How many read both?
6. Which of these is a set: (a) tall students in your class, (b) even numbers between 1 and 9?

Answers: 1. {1, 2, 3, 6, 9, 18} · 2. 2⁴ − 1 = 15 · 3. A ∩ B = {3, 5, 7}; A ∪ B = {1, 2, 3, 5, 7, 9}, so (A ∪ B)′ = {4, 6, 8, 10} · 4. 15 + 12 − 20 = 7 · 5. 60 − 10 = 50 read at least one, so both = 35 + 30 − 50 = 15 · 6. (b) only: "tall" is not well-defined

Next class: Binary Operations.