8 min readConcept sheetShort answer on notation; a 10-mark Venn-diagram survey problem

Set theory and Venn diagrams

A set is a well-defined collection. Union (∪) joins sets; intersection (∩) keeps only what they share. Venn diagrams make survey counts easy.

Membershipx ∈ A means x is in A

Cardinalityn(A) = number of elements

Key formulan(A ∪ B) = n(A) + n(B) − n(A ∩ B)

Subsets of an n-set2ⁿ

Set words and symbols

Idea	Symbol	Meaning	Example (A = {1, 2, 3})
Universal set	U	Everything under discussion	U = {1, 2, …, 10}
Empty set	∅ or { }	A set with no elements	{months with 32 days} = ∅
Subset	B ⊆ A	Every element of B is in A	{1, 2} ⊆ A
Union	A ∪ B	In A or B or both	A ∪ {3, 4} = {1, 2, 3, 4}
Intersection	A ∩ B	In both A and B	A ∩ {3, 4} = {3}
Complement	A′	In U but not in A	A′ = {4, 5, …, 10}
Difference	A − B	In A but not in B	A − {3, 4} = {1, 2}
Cardinality	n(A)	Number of elements	n(A) = 3
Set-builder	{x \| rule}	Describe by a rule	{x ∈ U \| x is even, x < 10} = {2, 4, 6, 8}

Counting formulas

Two setsn(A ∪ B) = n(A) + n(B) − n(A ∩ B)

Three setsn(A ∪ B ∪ C) = n(A) + n(B) + n(C) − n(A ∩ B) − n(A ∩ C) − n(B ∩ C) + n(A ∩ B ∩ C)

Only A (two sets)n(A) − n(A ∩ B)

Complementn(A′) = n(U) − n(A)

Number of subsets of an n-element set2ⁿ

De Morgan's laws(A ∪ B)′ = A′ ∩ B′ and (A ∩ B)′ = A′ ∪ B′

Set operations

Problem

U = {1, 2, …, 10}, A = {2, 4, 6, 8, 10}, B = {1, 2, 3, 4, 5}. Find A ∪ B, A ∩ B, A′, A − B and n(A ∪ B). Check De Morgan's law (A ∪ B)′ = A′ ∩ B′.

Working

A ∪ B: everything in either set = {1, 2, 3, 4, 5, 6, 8, 10}.
A ∩ B: only the shared elements = {2, 4}.
A′: elements of U not in A = {1, 3, 5, 7, 9}.
A − B: in A but not in B = {6, 8, 10}.
n(A ∪ B) = n(A) + n(B) − n(A ∩ B) = 5 + 5 − 2 = 8. Counting the list also gives 8. ✓
(A ∪ B)′ = {7, 9}. B′ = {6, 7, 8, 9, 10}, so A′ ∩ B′ = {7, 9}. ✓

Answer

A ∪ B = {1, 2, 3, 4, 5, 6, 8, 10}; A ∩ B = {2, 4}; A′ = {1, 3, 5, 7, 9}; A − B = {6, 8, 10}; n(A ∪ B) = 8.

A two-set customer survey

Problem

A Colombo supermarket surveyed 120 customers: 70 buy tea, 55 buy coffee and 25 buy both. How many buy (a) at least one, (b) tea only, (c) coffee only, (d) neither?

Working

Draw two overlapping circles, T and C, inside a rectangle U (120). Start in the middle: T ∩ C = 25.
(a) n(T ∪ C) = 70 + 55 − 25 = 100.
(b) Tea only = 70 − 25 = 45.
(c) Coffee only = 55 − 25 = 30.
(d) Neither = 120 − 100 = 20.
Check: 45 + 25 + 30 + 20 = 120. ✓

Answer

(a) 100 (b) 45 (c) 30 (d) 20

A three-set survey

Problem

Of 200 BBM students, 90 study Accounting (A), 80 Economics (E) and 70 Maths (M). 30 take A and E, 25 take A and M, 20 take E and M, and 10 take all three. Find (a) the number taking at least one, (b) none, (c) Accounting only.

Working

Fill the Venn diagram from the centre out. All three = 10.
A and E only = 30 − 10 = 20; A and M only = 25 − 10 = 15; E and M only = 20 − 10 = 10.
A only = 90 − 20 − 15 − 10 = 45; E only = 80 − 20 − 10 − 10 = 40; M only = 70 − 15 − 10 − 10 = 35.
(a) Formula: 90 + 80 + 70 − 30 − 25 − 20 + 10 = 175. Regions: 45 + 40 + 35 + 20 + 15 + 10 + 10 = 175. ✓
(b) None = 200 − 175 = 25.
(c) Accounting only = 45.

Answer

(a) 175 (b) 25 (c) 45

In Sri Lankan business

Market researchers in Sri Lanka can show brand overlap with Venn diagrams, e.g. households that use both Dialog and another mobile network. The overlap is counted once, which is exactly the formula n(A ∪ B) = n(A) + n(B) − n(A ∩ B).

Exam angle

Short answer: “If n(A) = 25, n(B) = 18 and n(A ∩ B) = 7, find n(A ∪ B).” 25 + 18 − 7 = 36.

Short answer: “How many subsets does {a, b, c, d} have?” 2⁴ = 16, including ∅ and the set itself.

Essay: “200 students: 90 Accounting, 80 Economics, 70 Maths; 30 A and E, 25 A and M, 20 E and M; 10 all three. Draw a Venn diagram and find how many take none and Accounting only.” Union = 175, none = 25, Accounting only = 45.

Remember it

Fill a Venn diagram from the centre outwards: all three first, then the pairs (subtract the centre), then the singles. Add every region to check the total.