Sets, Set Notation & Venn Diagrams
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Lesson overview
Types of set, union, intersection and complement, the laws of set algebra, and solving two- and three-set problems with Venn diagrams.
Go to lesson quiz →Exam guidance · JAMB UTME Speed Lens
Pacing
~40 seconds per MCQ (180 questions in 120 minutes, no negative marking).
How JAMB marks this
Fast dimensional/sign elimination and standard formula shortcuts — skip multi-line algebra when substitution tests options faster.
Watch out for
UTME distractors routinely include the intermediate value (e.g., forgetting to convert cm³ to dm³ or °C to Kelvin).
Lesson content
Overview
A school wants to know how many SS3 students offer both Physics and Chemistry. A survey of Lagos commuters asks who uses BRT buses, who uses danfos, and who uses neither. Questions like these are about sets: collections of objects sorted by a clear rule. JAMB tests sets in two ways. Short questions check notation and vocabulary: subsets, complements and the number of subsets. Longer questions give survey numbers and expect you to fill in a Venn diagram and read off a missing region. Both reward neat, systematic working.
Key concept: sets and their types
A set is a well-defined collection of objects, called elements or members. We write x ∈ A for 'x is an element of A' and x ∉ A for 'x is not an element of A'. n(A) is the number of elements in A. The universal set U contains every element under discussion. The empty (null) set ∅ or { } has no elements. A is a subset of B (A ⊆ B) if every element of A is also in B. A proper subset (A ⊂ B) is a subset that is not equal to B. A set with n elements has 2ⁿ subsets and 2ⁿ − 1 proper subsets. Equal sets have exactly the same elements. Equivalent sets have the same number of elements. Disjoint sets have no element in common. Union A ∪ B: elements in A or B or both. Intersection A ∩ B: elements in both A and B. Complement A′: elements of U that are not in A.
Set algebra and counting regions
The operations obey laws much like those of ordinary 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′. When you add n(A) and n(B), the elements in A ∩ B are counted twice, so one copy must be removed. That gives the key counting rule n(A ∪ B) = n(A) + n(B) − n(A ∩ B). To fill a Venn diagram, always start in the middle and work outwards. Put the 'both' (or 'all three') number in first. Subtract it from each set total to get the 'only' regions. Finally, subtract everything inside the circles from n(U) to get the region outside. If the 'both' number is unknown, call it x, write every region in terms of x, and form an equation using the total.
Formulae to know
Two sets: n(A ∪ B) = n(A) + n(B) − n(A ∩ B) With a universal set: n(U) = n(A ∪ B) + n((A ∪ B)′) Only A: n(A only) = n(A) − n(A ∩ B) Three sets: n(A ∪ B ∪ C) = n(A) + n(B) + n(C) − n(A ∩ B) − n(A ∩ C) − n(B ∩ C) + n(A ∩ B ∩ C) Subsets of a set with n elements: 2ⁿ (proper subsets: 2ⁿ − 1)
Worked example 1: listing elements
U = {1, 2, 3, …, 10}, A = {multiples of 2} and B = {multiples of 3}. Find A ∩ B, A ∪ B and A′ ∩ B.
Active Worked Example · Predict Step 1
5 stepsWrite down the formula, rule, or first move you would make before revealing Step 1.
Worked example 2: a two-set survey
In a class of 50 students, 32 offer Physics, 28 offer Chemistry and 5 offer neither subject. How many offer both? How many offer Physics only?
Active Worked Example · Predict Step 1
5 stepsWrite down the formula, rule, or first move you would make before revealing Step 1.
Worked example 3 (UTME standard): three sets
Of 40 traders at a market, 20 sell rice (R), 18 sell beans (B) and 15 sell garri (G). 7 sell rice and beans, 6 sell rice and garri, 5 sell beans and garri, and 2 sell all three. How many sell none of the three, and how many sell rice only?
Active Worked Example · Predict Step 1
6 stepsWrite down the formula, rule, or first move you would make before revealing Step 1.
Common traps
• Writing the set totals straight into the 'only' regions. The 32 who offer Physics include those who offer both. • Forgetting the region outside the circles. If n(U) is given, the 'neither' group is usually what the question wants. • Mixing up ∪ and ∩. Union is 'or' (everything); intersection is 'and' (only the overlap). • Counting the empty set wrongly. ∅ is a subset of every set, and every set is a subset of itself. • In three-set questions, subtracting the 'all three' value only once. Each pairwise overlap must have the centre removed before you find the 'only' regions.
Summary
• A set is a well-defined collection; know U, ∅, subset, proper subset, equal, equivalent and disjoint sets. • A ∪ B is 'or', A ∩ B is 'and', and A′ is 'not in A'. De Morgan: (A ∪ B)′ = A′ ∩ B′ and (A ∩ B)′ = A′ ∪ B′. • n(A ∪ B) = n(A) + n(B) − n(A ∩ B); extend it to three sets with the inclusion–exclusion formula. • Fill Venn diagrams from the centre outwards and check that all regions add up to n(U). • A set with n elements has 2ⁿ subsets and 2ⁿ − 1 proper subsets.
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