Mathematics notes, chapter 11

Introduction of set

Textbook Mathematics Form 1 (DLP), pages 246–262

What this chapter covers

Explain the meaning of a set by grouping objects that share a common characteristic (11.1)
Describe a set by description, by listing and by set builder notation (11.1)
Recognise an empty set and write it as φ or { } (11.1)
Say whether an object is an element of a set and write the relation with ∈ or ∉ (11.1)
Find the number of elements of a set and write it as n(A) (11.1)
Decide whether two or more sets are equal and explain why (11.1)
Identify and describe the universal set ξ and the complement A′ of a set (11.2)
Represent a set, the universal set and the complement of a set with Venn diagrams (11.2)
Identify subsets (⊂, ⊄) and list all the possible subsets of a set, using 2ⁿ to count them (11.2)
Represent subsets and the relations between sets, subsets, universal sets and complements with Venn diagrams (11.2)

At home we sort rubbish for recycling into paper, plastic, glass and aluminium. Each group holds objects that share one characteristic. In Mathematics such a group is called a set. In this chapter you will learn to write sets in three ways, count their elements, compare them, and draw them as Venn diagrams.

Set theory was first introduced by the German mathematician Georg Cantor (1845–1918) in 1874. Venn diagrams are named after the English mathematician John Venn (1834–1923).

English	Bahasa Melayu
set	set
element	unsur
description, set notation, set builder notation	perihalan, tatatanda set, tatatanda pembina set
empty set	set kosong
equal sets	set sama
universal set	set semesta
complement of a set	pelengkap bagi suatu set
subset	subset
Venn diagram	gambar rajah Venn

11.1Set

What is a set?

When we sort objects, the objects with a common characteristic go into the same group. For example, car, lorry, van and bus all travel on land.

A set is a group of objects with a common characteristic. Each object in the set is called an element. Sets are named with capital letters such as A, P and Q.

Classify these as land, water or air transport: car, rocket, sampan, lorry, aeroplane, boat, ship, van, ferry, helicopter, bus, hot-air balloon.

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Land: car, lorry, van, bus
Water: sampan, boat, ship, ferry
Air: rocket, aeroplane, helicopter, hot-air balloon

Three ways to describe a set

Method	How	The colours of the Jalur Gemilang
Description	A sentence in words	A is the set of colours of the Jalur Gemilang.
Listing (set notation)	Write the elements inside { }, separated by commas	A = {red, white, blue, yellow}
Set builder notation	{x : the condition x must satisfy}	A = {x : x is a colour of the Jalur Gemilang}

Read {x : …} as "the set of all x such that …". The colon ":" means "such that".

Example 1 (textbook p. 249)

Describe each set by description, listing and set builder notation. (a) Letters in the word 'MALAYSIA' (b) Odd numbers which are less than 20

(a) Description: P is the set of letters in the word 'MALAYSIA'.
Listing: P = {M, A, L, Y, S, I}. The letter A appears three times in the word, but it is written only once.
Set builder notation: P = {x : x is a letter in the word 'MALAYSIA'}
(b) Description: Q is the set of odd numbers which are less than 20.
Listing: Q = {1, 3, 5, 7, 9, 11, 13, 15, 17, 19}
Set builder notation: Q = {x : x is an odd number and x < 20}

Do not repeat an element when you list a set. The letters in 'KELANTAN' are {K, E, L, A, N, T}, not {K, E, L, A, N, T, A, N}.

The empty set

Which month has 32 days? None! The set M of months with 32 days has no elements.

A set with no elements is an empty set (or null set). It is written φ (read "phi") or { }. So M = φ or M = { }.

{0} is not an empty set. It has one element, the number 0. Also, never write {φ}: that is a set containing one element.

TRUE or FALSE? a) If P = {triangles which have diagonals}, then P = φ. b) If Q = {prime numbers which are also even numbers}, then Q = { }. c) If R = {odd numbers which are divisible by 2}, then R = φ.

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a) TRUE: a triangle has no diagonals.
b) FALSE: 2 is a prime number and an even number, so Q = {2}.
c) TRUE: no odd number is divisible by 2.

Elements: ∈ and ∉

Let A = {local fruits in Malaysia}. Durian is a local fruit, so durian is an element of A. Apple is not.

Symbol	Meaning	Example
∈ (epsilon)	is an element of	durian ∈ A
∉	is not an element of	apple ∉ A

Example 2 (textbook p. 251)

Given P = {x : x is a prime number and 0 ≤ x ≤ 20}, complete with ∈ or ∉: (a) 5 __ P (b) 8 __ P (c) 19 __ P (d) 1 __ P

First list all the elements: P = {2, 3, 5, 7, 11, 13, 17, 19}
(a) 5 ∈ P (b) 8 ∉ P (c) 19 ∈ P (d) 1 ∉ P (1 is not a prime number)

When a set is given by description or set builder notation, list it first. Then checking ∈ or ∉ is easy.

P = {x : x is an odd number and 0 < x < 50}, Q = {x : x is a common multiple of 4 and 6, x < 100}. Complete with ∈ or ∉: a) 8 __ P b) 17 __ P c) 63 __ P d) 84 __ Q e) 60 __ Q f) 46 __ Q

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Q = {12, 24, 36, 48, 60, 72, 84, 96}
a) 8 ∉ P (even) b) 17 ∈ P c) 63 ∉ P (63 is odd but not less than 50)
d) 84 ∈ Q e) 60 ∈ Q f) 46 ∉ Q

The number of elements, n(A)

The number of elements in set P is written n(P). If P = {a, b, c, d, e, f}, then n(P) = 6.

Example 3 (textbook p. 252)

P = {letters in the word 'WAWASAN'}, Q = {x : x is a two-digit number such that the sum of its digits is 7}. Find n(P) and n(Q).

P = {W, A, S, N}, so n(P) = 4 (not 7: repeated letters are counted once).
Q = {16, 25, 34, 43, 52, 61, 70}, so n(Q) = 7. Do not forget 70: 7 + 0 = 7.

Find a) n(B), B = {consonants in the word 'KEMAHIRAN'} b) n(D), D = {multiples of 11 which are less than 100} c) n(P), P = {integers between −5 and 2}

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a) B = {K, M, H, R, N}, n(B) = 5
b) D = {11, 22, 33, …, 99}, n(D) = 9
c) P = {−4, −3, −2, −1, 0, 1}, n(P) = 6 ("between" does not include −5 and 2)

Equal sets

Three students list the letters of 'HARUM': A = {H, A, R, U, M}, B = {M, U, R, A, H}, C = {R, U, M, A, H}. The elements are the same, only the order is different.

Sets are equal if they have exactly the same elements. We write A = B = C. The order of the elements does not matter. If two sets are not equal we write A ≠ B.

Example 4 (textbook pp. 252–253)

Are these pairs equal sets? (a) P = {vowels in the word 'SEKOLAH'}, Q = {vowels in the word 'KEJOHANAN'} (b) F = {perfect squares which are less than 50}, G = {4, 9, 16, 25, 36, 49}

(a) P = {E, O, A} and Q = {E, O, A}. Every element is the same, so P = Q.
(b) F = {1, 4, 9, 16, 25, 36, 49}. 1 ∈ F but 1 ∉ G, so F ≠ G.

Equal sets always have the same number of elements, but the same number of elements does not mean equal sets. {1, 2, 3} and {a, b, c} both have 3 elements, but they are not equal.

Equal or not? a) G = {letters in 'RAJIN'}, H = {letters in 'JIRAN'} b) M = {1, 2, 3, 4, 5}, N = {whole numbers which are less than 6} c) P = {x : x is a multiple of 2 and 25 ≤ x ≤ 45}, Q = {x : x is a number that has the digit 2 and 25 ≤ x ≤ 45}

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a) Equal: both are {R, A, J, I, N}.
b) Not equal: N = {0, 1, 2, 3, 4, 5} has the element 0 (whole numbers start at 0).
c) Not equal: P = {26, 28, 30, …, 44} but Q = {25, 26, 27, 28, 29, 32, 42}; for example 30 ∈ P but 30 ∉ Q.

11.2Venn diagrams, universal sets, complement of a set and subsets

Universal set and complement of a set

Eight students took part in a Mathematics quiz: Amir, Hazura, Laila, Sandra, Zamri, Dali, Pei San and Yana. These are all the students under discussion. Five of them, Amir, Hazura, Laila, Sandra and Zamri, are members of the Mathematics Society (set A).

The universal set, ξ (read "xi"), contains all the elements under discussion. The complement of set A, written A′, is the set of elements of ξ that are not in A.

ξ = {Amir, Hazura, Laila, Sandra, Zamri, Dali, Pei San, Yana}, A = {Amir, Hazura, Laila, Sandra, Zamri}, so A′ = {Dali, Pei San, Yana}.

Example 5 (textbook p. 254)

Is each set a universal set of {2, 3, 5, 7}? (a) {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} (b) {odd numbers which are less than 10}

(a) It contains 2, 3, 5 and 7, so it is a universal set.
(b) {1, 3, 5, 7, 9} does not contain 2, so it is not a universal set.

Example 6 (textbook pp. 254–255)

ξ = {x : x is an integer and 1 ≤ x ≤ 10}. Find the complement of (a) P = {perfect squares which are less than 10} (b) Q = {factors of 10}.

ξ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
(a) P = {1, 4, 9}, so P′ = {2, 3, 5, 6, 7, 8, 10}
(b) Q = {1, 2, 5, 10}, so Q′ = {3, 4, 6, 7, 8, 9}

Check: n(P) + n(P′) = n(ξ). In Example 6(a), 3 + 7 = 10 ✓

ξ = {whole numbers which are less than 10}. Find the complement of a) P = {multiples of 3} b) Q = {prime numbers}.

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ξ = {0, 1, 2, …, 9}
a) P = {3, 6, 9}, P′ = {0, 1, 2, 4, 5, 7, 8} (like the textbook, the multiples of 3 here are 3, 6, 9, so 0 goes into P′)
b) Q = {2, 3, 5, 7}, Q′ = {0, 1, 4, 6, 8, 9}

Venn diagrams

A Venn diagram shows sets as closed shapes. The universal set is drawn as a rectangle. Each set inside it is usually a circle or an oval (a triangle or rectangle is fine too). Each element is a dot with its name.

Venn diagram: rectangle xi with all 8 students; oval A contains Amir, Hazura, Laila, Sandra and Zamri; Dali, Pei San and Yana are outside the oval — Elements of A are inside the oval; the elements of A′ lie outside it, inside the rectangle

Example 7 (textbook p. 256)

ξ = {x : 10 < x < 20, x is an integer}, M = {11, 17} and N = {odd numbers}. Draw a Venn diagram to show (a) ξ and M (b) N′.

ξ = {11, 12, 13, 14, 15, 16, 17, 18, 19}; N = {11, 13, 15, 17, 19}
Write 11 and 17 inside M, and the other 7 elements outside M.
For (b), write the odd numbers inside N and shade the region outside N: that region is N′ = {12, 14, 16, 18}.

Venn diagram: M contains 11 and 17; 12, 13, 14, 15, 16, 18, 19 outside M

Venn diagram: N contains 11, 13, 15, 17, 19; region outside N is shaded and contains 12, 14, 16, 18

To shade A′, shade everything inside the rectangle except set A. Don't shade only a small piece of the rectangle.

Subsets

Let A = {multiples of 2} and B = {multiples of 4}. Every multiple of 4 is also a multiple of 2, so every element of B is also in A.

B is a subset of A, written B ⊂ A, if every element of B is also an element of A. If at least one element of B is not in A, B is not a subset of A: B ⊄ A.

Example 8 (textbook p. 257)

Is A a subset of B? (a) A = {2, 4, 6}, B = {1, 2, 3, 4, 5, 6} (b) A = {prime numbers which are less than 20}, B = {odd numbers which are less than 20}

(a) 2, 4 and 6 are all in B, so A ⊂ B.
(b) A = {2, 3, 5, 7, 11, 13, 17, 19}. The element 2 is not odd, so 2 ∉ B and A ⊄ B.

The empty set φ is a subset of every set, and every set is a subset of itself. A set with n elements has 2ⁿ subsets.

Example 9 (textbook p. 258)

List all the possible subsets of (a) {3, 4} (b) {a, b, c}.

(a) φ, {3}, {4}, {3, 4}: that is 2² = 4 subsets.
(b) Start with φ, then one element, two elements, three elements:
φ, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}: 2³ = 8 subsets.

Number of elements, n	0	1	2	3	4	5
Number of subsets, 2ⁿ	1	2	4	8	16	32

Students often forget φ and the set itself when they list subsets. Count your list: it must have 2ⁿ subsets.

a) Complete with ⊂ or ⊄: A = {a, u}, B = {a, e, i, o, u}: A __ B. b) M = {odd numbers which are less than 50}, N = {multiples of 5 which are less than 50}: N __ M. c) List all the subsets of {p, q}. d) How many subsets does {prime numbers which are less than 10} have?

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a) A ⊂ B
b) N ⊄ M (for example 10 ∈ N but 10 is even, so 10 ∉ M)
c) φ, {p}, {q}, {p, q}
d) {2, 3, 5, 7} has 4 elements, so 2⁴ = 16 subsets

Subsets in Venn diagrams

If B ⊂ A, the shape for B is drawn completely inside the shape for A. Here A = {2, 4, 6, …, 20} and B = {4, 8, 12, 16, 20}.

Venn diagram: B with 4, 8, 12, 16, 20 drawn inside A, which also contains 2, 6, 10, 14, 18 — B ⊂ A: the oval B lies inside the oval A

Example 10 (textbook p. 259)

Draw Venn diagrams for (a) A = {p, q, r, s, t} and B = {p, r, s} (b) P = {whole numbers} and Q = {prime numbers}.

(a) Every element of B is in A, so B ⊂ A. Put p, r, s inside B, and q, t in A but outside B.
(b) Every prime number is a whole number, so Q ⊂ P. These sets are infinite, so the elements are not written.

B inside A; p, r, s in B; q and t in A only

Relationships between sets

One Venn diagram can show the universal set, subsets and complements together.

Example 11 (textbook p. 259)

ξ = {x : x ≤ 10, x is a positive integer}, A = {factors of 10}, B = {numbers which are divisible by 5}. Draw a Venn diagram.

List the sets: ξ = {1, 2, …, 10}, A = {1, 2, 5, 10}, B = {5, 10}
B ⊂ A, so draw B inside A, and A inside the rectangle ξ.
Write 5 and 10 in B; 1 and 2 in A outside B; 3, 4, 6, 7, 8, 9 outside A.

Venn diagram: xi contains A; B inside A contains 5 and 10; 1 and 2 in A only; 3, 4, 6, 7, 8, 9 outside A

Draw the biggest set first and work inwards. Then tick off each element of ξ as you write it, so none is missing or written twice.

Draw a Venn diagram for ξ = {a, b, c, d, e, f, g, h}, P = {a, b, c, d}, Q = {b}, R = {f, g}.

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Q ⊂ P, so Q is drawn inside P. R has no element in common with P, so R is drawn separately. e and h are outside all the sets.
Venn diagram: Q = {b} inside P with a, c, d; R separate with f, g; e and h outside

Venn diagram: Q = {b} inside P with a, c, d; R separate with f, g; e and h outside

Summary

Idea	Key fact	Example
Set	A group of objects with a common characteristic	A = {red, white, blue, yellow}
Describing a set	Description, listing, set builder notation	Q = {x : x is an odd number and x < 20}
Empty set	No elements: φ or { }	{months with 32 days} = φ
Element	∈ is an element of, ∉ is not	5 ∈ {primes}, 1 ∉ {primes}
Number of elements	n(A); count repeated letters once	n({W, A, S, N}) = 4
Equal sets	Exactly the same elements, any order	{S, A, M} = {M, A, S}
Universal set ξ	All the elements under discussion; a rectangle	ξ = {1, 2, …, 10}
Complement A′	Elements of ξ not in A; n(A) + n(A′) = n(ξ)	A = {1, 4, 9}, A′ = {2, 3, 5, 6, 7, 8, 10}
Subset	B ⊂ A if every element of B is in A; 2ⁿ subsets	{a, b, c} has 2³ = 8 subsets

Rectangle xi with oval P; the region outside P is shaded — Shaded region is P′

Oval R drawn inside oval Q inside rectangle xi — Shaded region is P′