Mathematics · Lesson 5

Sets & Venn diagrams

JAMBWAECNECO

What the syllabus asks

JAMB: Sets: types of sets, algebra of sets, Venn diagrams and their applications

WAEC: Sets: idea of sets, universal set, finite and infinite sets, subset, empty set, disjoint sets, equal and equivalent sets, power set; set notation; union, intersection and complement; Venn diagrams and their use in solving problems with two and three sets

NECO: Sets: idea of sets, universal set, finite and infinite sets, subset, empty set, disjoint sets, equal and equivalent sets, power set; set notation; union, intersection and complement; Venn diagrams and their use in solving problems with two and three sets

What you will learn

Describe sets by listing, by rule and in set-builder notation.
Identify types of sets: finite, infinite, empty, equal, equivalent, disjoint, universal.
Find subsets and the number of subsets of a set.
Find unions, intersections and complements.
Draw and interpret Venn diagrams for two and three sets.
Solve word problems with Venn diagrams and the formula n(A ∪ B) = n(A) + n(B) − n(A ∩ B).

Key points

A set is a well-defined collection; elements ∈; n(A) counts them.
Types: finite, infinite, empty ∅, unit, equal, equivalent, disjoint, universal.
n elements → 2ⁿ subsets and 2ⁿ − 1 proper subsets.
Union ∪ (or), intersection ∩ (and), complement ′ (not).
n(A ∪ B) = n(A) + n(B) − n(A ∩ B).
Three sets: fill from the centre outward; check regions sum to n(U).
De Morgan: (A ∪ B)′ = A′ ∩ B′; (A ∩ B)′ = A′ ∪ B′.

Start here: What is a set?

A set is a well-defined collection of objects. "Well-defined" means we can always decide whether an object belongs. "The days of the week" is a set. "The best musicians in Nigeria" is not a well-defined set, because people disagree on who is "best".

The objects in a set are its elements or members. We use capital letters for sets and curly brackets for their elements:

A = {Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday}.

We write ∈ for "is an element of" and ∉ for "is not an element of": Monday ∈ A, January ∉ A. The number of elements is written n(A); here n(A) = 7.

Ways of describing a set

Listing (roster): B = {2, 4, 6, 8}.
Description in words: B is the set of even numbers between 1 and 9.
Set-builder notation: B = {x : x is even, 1 < x < 9}. Read "the set of x such that x is even and between 1 and 9".

Worked example 5.1

List the elements of P = {x : x is prime, 2 < x ≤ 9}.

Numbers from 3 to 9: 3, 4, 5, 6, 7, 8, 9.
Primes among them: 3, 5, 7. (2 is excluded because x > 2.)

P = {3, 5, 7}

Try it yourself. List Q = {x : x² = 16}.

Show solution

{−4, 4}.

Sets are everywhere once you look: the set of states in Nigeria (n = 36), the set of subjects you will write in WAEC, the set of vowels {a, e, i, o, u}. Thinking in sets helps you sort information clearly.

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Practice questions

1. List {x : x is a factor of 20}.

Show worked answer

{1, 2, 4, 5, 10, 20}.

2. How many proper subsets has {p, q, r, s}?

Show worked answer

2⁴ − 1 = 15.

3. U = {1,…,10}, A = {odd numbers}, B = {primes}. Find A ∩ B.

Show worked answer

{3, 5, 7}.

4. n(U) = 45, n(A) = 25, n(B) = 22, n(A ∩ B) = 9. Find n(A ∪ B)′.

Show worked answer

45 − 38 = 7.

5. In a class of 42, 26 like music, 20 like art, 6 like neither. How many like both?

Show worked answer

26 + 20 − 36 = 10.

Key words

Set: A well-defined collection of objects.
Element: A member of a set (∈).
Universal set (U or ξ): The set of all elements under consideration.
Empty (null) set: A set with no elements, ∅ or { }.
Finite set: A set with a countable number of elements.
Infinite set: A set whose elements cannot be counted to an end.
Subset: A set whose elements all belong to another set.
Proper subset: A subset not equal to the whole set.
Power set: The set of all subsets of a set.
Equal sets: Sets with exactly the same elements.
Equivalent sets: Sets with the same number of elements.
Disjoint sets: Sets with no common elements.

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