Mathematics notes, chapter 4

Operations on Sets

Textbook Mathematics Form 4 KSSM (DLP), pages 94–127

What this chapter covers

Determine and describe the intersection of sets using various representations, and write it with the symbol ∩ (4.1)
Determine the complement of the intersection of sets, list its elements and shade it on a Venn diagram (4.1)
Solve problems involving the intersection of sets (4.1)
Determine and describe the union of sets using various representations, and write it with the symbol ∪ (4.2)
Determine the complement of the union of sets (4.2)
Solve problems involving the union of sets, using n(A ∪ B) = n(A) + n(B) − n(A ∩ B) (4.2)
Determine and describe combined operations on sets, such as (P ∪ Q) ∩ R, and their complements, with brackets worked first (4.3)
Solve problems involving combined operations on sets, using Venn diagrams with two or three sets (4.3)

Sets are used to sort and compare data: which pupils join the football club, the drama club, or both? In Form 1 you listed sets and drew simple Venn diagrams. In this chapter you will combine sets with three operations, intersection (∩), union (∪) and complement (′), and use Venn diagrams with two or three sets to solve problems with numbers.

Set theory was pioneered by the German mathematician Georg Cantor (1845–1918). Venn diagrams with sets inside a rectangle ξ show all the elements at once, so they are the main tool of this chapter.

English	Bahasa Melayu
intersection	persilangan
union	kesatuan
complement	pelengkap
set, empty set, subset	set, set kosong, subset
element, set notation	unsur, tatatanda set

4.1Intersection of sets

What is an intersection?

Some pupils like durian, some like rambutan, and a few like both. The pupils who like both fruits belong to the set A ∩ B.

The intersection of sets P and Q, written P ∩ Q, is the set of elements that are in both P and Q. Read "∩" as "and".

Two overlapping ovals P and Q inside a rectangle; only the overlap is shaded — P ∩ Q: the overlap

Two overlapping ovals P and Q inside a rectangle; everything except the overlap is shaded — P ∩ Q: the overlap

Example 1 (like textbook p. 98)

ξ = {x : x is an integer, 1 ≤ x ≤ 10}, P = {odd numbers}, Q = {prime numbers} and R = {multiples of 3}. (a) List P ∩ Q, P ∩ R, Q ∩ R and P ∩ Q ∩ R. (b) State n(P ∩ Q).

List each set first: P = {1, 3, 5, 7, 9}, Q = {2, 3, 5, 7}, R = {3, 6, 9}.
(a) P ∩ Q = {3, 5, 7}; P ∩ R = {3, 9}; Q ∩ R = {3}; P ∩ Q ∩ R = {3} (only 3 is in all three sets).
(b) n(P ∩ Q) = 3.

Venn diagram of xi = 1 to 10 with P, Q and R; the overlap of P and Q is shaded and contains 3, 5 and 7 — P ∩ Q is shaded. Element 3 is in P, Q and R.

An element in all three sets lies in the middle of the Venn diagram. Put it there first, then fill the regions where only two sets overlap, then the sets alone.

When there is nothing in common

Example 2 (like textbook p. 99)

A = {numbers on a dice}, B = {even numbers on a dice}, C = {7, 8, 9}. Find A ∩ B, B ∩ C and A ∩ C.

A = {1, 2, 3, 4, 5, 6}, B = {2, 4, 6}.
A ∩ B = {2, 4, 6} = B, because B ⊂ A.
B ∩ C = { } and A ∩ C = φ: they share no element. The Venn diagram draws B and C apart.

If B ⊂ A then A ∩ B = B (the smaller set), not A. And if two sets have no common element, the answer is the empty set φ, not "0".

ξ = {x : x is an integer, 1 ≤ x ≤ 10}, M = {odd numbers}, N = {multiples of 3}. A = {letters in KELAS}, B = {letters in SEKOLAH}, C = {letters in LAUT}. a) List M ∩ N. b) Find n(A ∩ B). c) List A ∩ C.

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a) M = {1, 3, 5, 7, 9}, N = {3, 6, 9}, so M ∩ N = {3, 9}
b) A = {K, E, L, A, S}, B = {S, E, K, O, L, A, H}. A ∩ B = {K, E, L, A, S} = A, so n(A ∩ B) = 5
c) C = {L, A, U, T}, so A ∩ C = {L, A}

Complement of an intersection

(P ∩ Q)′ is the set of all elements of ξ that are not in P ∩ Q. Work in two steps: find P ∩ Q, then take everything else in ξ.

Example 3 (like textbook p. 100)

ξ = {1, 2, …, 10}, A = {2, 4, 6, 8, 10} and B = {3, 6, 9}. List (A ∩ B)′ and find n((A ∩ B)′).

A ∩ B = {6}.
(A ∩ B)′ = {1, 2, 3, 4, 5, 7, 8, 9, 10}, so n((A ∩ B)′) = 9.
Check: n(A ∩ B) + n((A ∩ B)′) = 1 + 9 = 10 = n(ξ).

Venn diagram of xi = 1 to 10 with ovals A and B overlapping at 6; everything outside the overlap is shaded — (A ∩ B)′: shade everything except the overlap, including the elements outside both ovals

(A ∩ B)′ is not the same as A′ ∩ B′. Here A′ ∩ B′ would leave out all of A and all of B.

ξ = {1, 2, …, 8}, A = {1, 2, 3, 4, 5, 6}, B = {2, 4, 6}. List (A ∩ B)′ and state its number of elements.

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A ∩ B = {2, 4, 6} (because B ⊂ A). So (A ∩ B)′ = {1, 3, 5, 7, 8} and the number of elements is 5.

4.2Union of sets

What is a union?

The union of sets P and Q, written P ∪ Q, is the set of elements that are in P or in Q or in both. Read "∪" as "or". Each element is written once only.

Two overlapping ovals P and Q inside a rectangle; both ovals are fully shaded — P ∪ Q: both ovals

Two overlapping ovals P and Q inside a rectangle; only the area outside both ovals is shaded — P ∪ Q: both ovals

Example 4 (like textbook p. 108)

P = {factors of 12} and Q = {multiples of 3 which are less than 16}. List P ∪ Q and find n(P ∪ Q).

P = {1, 2, 3, 4, 6, 12}, Q = {3, 6, 9, 12, 15}.
P ∪ Q = {1, 2, 3, 4, 6, 9, 12, 15}: the common elements 3, 6 and 12 are written once.
n(P ∪ Q) = 8.

Venn diagram: P holds 1, 2, 4 alone; the overlap holds 3, 6, 12; Q holds 9 and 15 alone; the whole of both ovals is shaded

For two sets: n(P ∪ Q) = n(P) + n(Q) − n(P ∩ Q). Adding n(P) and n(Q) counts the common elements twice, so subtract them once. Check: 6 + 5 − 3 = 8 ✓

Do not write 3, 6 and 12 twice when you list P ∪ Q. A union with repeated elements has the wrong n.

Complement of a union

(P ∪ Q)′ is the set of elements of ξ that are in neither P nor Q. Find P ∪ Q first, then list what is left in ξ.

Example 5 (like textbook p. 110)

ξ = {x : x is an integer, 20 ≤ x ≤ 30}, G = {prime numbers}, H = {multiples of 4}, I = {multiples of 5}. List (G ∪ H)′, (H ∪ I)′ and (G ∪ H ∪ I)′.

ξ = {20, 21, …, 30}, G = {23, 29}, H = {20, 24, 28}, I = {20, 25, 30}.
G ∪ H = {20, 23, 24, 28, 29}, so (G ∪ H)′ = {21, 22, 25, 26, 27, 30}, n = 6.
H ∪ I = {20, 24, 25, 28, 30}, so (H ∪ I)′ = {21, 22, 23, 26, 27, 29}, n = 6.
G ∪ H ∪ I = {20, 23, 24, 25, 28, 29, 30}, so (G ∪ H ∪ I)′ = {21, 22, 26, 27}, n = 4.

Venn diagram of xi = 20 to 30 with G, H and I; the area outside all three ovals is shaded and holds 21, 22, 26, 27 — (G ∪ H ∪ I)′ is the shaded region

a) A = {a, e, i}, B = {b, e, u}, C = {c, i, u, e}. List A ∪ B and A ∪ B ∪ C. b) ξ = {1, 2, …, 12}, X = {multiples of 4}, Y = {factors of 12}. List (X ∪ Y)′.

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a) A ∪ B = {a, b, e, i, u}; A ∪ B ∪ C = {a, b, c, e, i, u}
b) X = {4, 8, 12}, Y = {1, 2, 3, 4, 6, 12}, X ∪ Y = {1, 2, 3, 4, 6, 8, 12}, so (X ∪ Y)′ = {5, 7, 9, 10, 11}

Problems with two sets

Use the steps of the textbook: understand the problem, plan a strategy (draw a Venn diagram), carry it out, and check. Start filling from the middle (the intersection) outwards.

Example 6 (like textbook p. 113)

Of 60 pupils, 38 swim, 29 cycle and 12 do both. How many (a) swim only, (b) cycle only, (c) do neither?

Put 12 in the overlap.
(a) Swim only = 38 − 12 = 26. (b) Cycle only = 29 − 12 = 17.
n(S ∪ C) = 26 + 12 + 17 = 55, which agrees with 38 + 29 − 12 = 55.
(c) Neither = 60 − 55 = 5.

Venn diagram: swimming only 26, both 12, cycling only 17, outside both 5

38 is the number who swim in total, including the 12 who also cycle. Do not write 38 in the "swim only" part.

Problems with three sets

Example 7 (like textbook p. 114)

In a group of 90 learners, 30 join Art Club, 27 join Music Club and 33 join Sports Club. 9 join Art and Music, 8 join Music and Sports, 11 join Art and Sports, and 5 join all three clubs. How many join no club?

Start in the centre: all three = 5.
Exactly two clubs: Art and Music only = 9 − 5 = 4; Music and Sports only = 8 − 5 = 3; Art and Sports only = 11 − 5 = 6.
One club only: Art = 30 − 4 − 6 − 5 = 15; Music = 27 − 4 − 3 − 5 = 15; Sports = 33 − 3 − 6 − 5 = 19.
In at least one club: 15 + 15 + 19 + 4 + 3 + 6 + 5 = 67. No club: 90 − 67 = 23.

Three-set Venn diagram: Art only 15, Music only 15, Sports only 19, Art and Music 4, Music and Sports 3, Art and Sports 6, all three 5, none 23

Always write the intersection of all three first, then subtract it out when you fill the regions where two sets overlap.

a) n(ξ) = 40 pupils, n(K) = 15, n(L) = 9 and n(K ∩ L) = 5. How many are in neither set? b) Of 50 learners, 31 study Chinese and 28 study Tamil, and 5 study neither. How many study both?

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a) n(K ∪ L) = 15 + 9 − 5 = 19, so neither = 40 − 19 = 21
b) n(C ∪ T) = 50 − 5 = 45, and 31 + 28 − n(both) = 45, so both = 14

4.3Combined operations on sets

Intersection and union together

Combined operations use ∩, ∪ and ′ in one expression. Brackets first, and otherwise work from left to right. The complement ′ is applied to the set or bracket just before it.

Expression	Read as	Steps
(A ∪ B) ∩ C	elements in A or B, that are also in C	A ∪ B, then ∩ C
A′ ∩ (B ∪ C)	not in A, but in B or C	A′ and B ∪ C, then ∩
(A ∩ C)′ ∪ (B ∩ C)	not in A ∩ C, or in B ∩ C	(A ∩ C)′ and B ∩ C, then ∪

Example 8 (like textbook pp. 116–119)

ξ = {1, 2, …, 12}, A = {factors of 12}, B = {even numbers}, C = {multiples of 3}. List (a) (A ∪ B) ∩ C (b) A′ ∩ (B ∪ C).

A = {1, 2, 3, 4, 6, 12}, B = {2, 4, 6, 8, 10, 12}, C = {3, 6, 9, 12}.
(a) A ∪ B = {1, 2, 3, 4, 6, 8, 10, 12}. Of these, the ones in C: (A ∪ B) ∩ C = {3, 6, 12}.
(b) A′ = {5, 7, 8, 9, 10, 11}. B ∪ C = {2, 3, 4, 6, 8, 9, 10, 12}. A′ ∩ (B ∪ C) = {8, 9, 10}.

Three-set Venn diagram with xi = 1 to 12; the parts of C that lie in A or B are shaded: 3, 6, 12 — (A ∪ B) ∩ C

Three-set Venn diagram with xi = 1 to 12; the parts of B or C that lie outside A are shaded: 8, 10, 9 — (A ∪ B) ∩ C

Shading a combined operation

Shade each part on its own (in pencil, or in your head), then keep the part that meets the operation: for ∩ keep only the region shaded by both parts, for ∪ keep every region shaded by either part.

Three-set Venn diagram of P, Q and R; the parts of R that lie inside P or Q are shaded — (P ∪ Q) ∩ R: the part of R that is also in P or Q

Example 9 (like textbook p. 118)

Using the sets of Example 8, list (A ∩ C)′ ∪ (B ∩ C).

A ∩ C = {3, 6, 12}, so (A ∩ C)′ = {1, 2, 4, 5, 7, 8, 9, 10, 11}.
B ∩ C = {6, 12}.
Union: {1, 2, 4, 5, 6, 7, 8, 9, 10, 11, 12}. Only 3 is missing: it is in A ∩ C but not in B ∩ C.

(A ∪ B) ∩ C and A ∪ (B ∩ C) are different. In Example 8, A ∪ (B ∩ C) = {1, 2, 3, 4, 6, 12} ∪ {6, 12} = {1, 2, 3, 4, 6, 12}, but (A ∪ B) ∩ C = {3, 6, 12}. Brackets change the answer.

ξ = {1, 2, …, 10}, K = {1, 2, 3, 4, 5, 6}, L = {2, 4, 6, 8, 10}, M = {3, 6, 9}. List a) (K ∩ L) ∪ M b) K′ ∩ L c) (K ∪ M)′.

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a) K ∩ L = {2, 4, 6}; (K ∩ L) ∪ M = {2, 3, 4, 6, 9}
b) K′ = {7, 8, 9, 10}; K′ ∩ L = {8, 10}
c) K ∪ M = {1, 2, 3, 4, 5, 6, 9}; (K ∪ M)′ = {7, 8, 10}

Summary

Idea	Key fact	Example
Intersection P ∩ Q	In both P and Q	{1, 3, 5} ∩ {3, 5, 7} = {3, 5}
Union P ∪ Q	In P or Q or both; each element once	{1, 3, 5} ∪ {3, 5, 7} = {1, 3, 5, 7}
Complement of a result	(P ∩ Q)′ and (P ∪ Q)′: find the inside first, then take the rest of ξ	n(P ∩ Q) + n((P ∩ Q)′) = n(ξ)
Counting two sets	n(P ∪ Q) = n(P) + n(Q) − n(P ∩ Q)	38 + 29 − 12 = 55
Combined operations	Brackets first, then left to right	(A ∪ B) ∩ C can differ from A ∪ (B ∩ C)
Venn diagram problems	Fill from the middle outwards; check the total equals n(ξ)	15 + 15 + 19 + 4 + 3 + 6 + 5 + 23 = 90

Notes and examples follow Mathematics Form 4 KSSM (DLP), Chapter 4, pp. 94–127. Venn diagrams drawn for this site.