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C h a p t e r 1 0 E x t e n s i o n — S e t s a n d V e n n d i a g r a m s 1

Sets and Venn diagrams A set is a collection of objects and is usually referred to by a capital letter. The objects are the elements — or members — of the set and consist of numbers or any symbolic representation of the given information.

The elements of a set are separated by a comma and are enclosed by { }. For example, the set A consisting of the vowels of the English alphabet can be

written as A = {a, e, i, o, u}. The order of listing the elements is unimportant so we could also write A = {e, i, u, a, o} or A = {u, e, a, o, i} and so on.

To indicate that an element belongs to a set we use the symbol ∈. x ∈ A is read as ‘x is an element of set A’ or ‘x belongs to set A’. Similarly, x ∉ A is used to denote that an element does not belong to set A. For example, if W = {Monday, Tuesday} we can write Monday ∈ W and

Wednesday ∉ W.

Finite and infinite sets If all the elements of a set can be listed, the set is finite. An infinite set has elements that cannot all be listed. The elements are expressed descriptively or by listing the set’s first few elements and using dots to represent the remaining members of the set.

For example, the set N of all positive numbers can be described as: N = {positive even numbers} or N = {2, 4, 6, 8, . . .}

Cardinal number The cardinal number of a set A is the number of elements in A and denoted by n(A). For example, if A = {3, 5, 7, 9} then n(A) = 4 and if B = {letters of the English alphabet} then n(B) = 26.

Null set and unit set A set containing no elements is called a null set or empty set and is denoted by φ.

If the set consists of only one element, it is termed a unit set. For example, the set A = {positive numbers less than 0} can also be written as A = φ

and the set B = {even numbers from 3 to 5 inclusive} is a unit set which can also be described as B = {4}.

List the elements of the set A = {odd numbers from 3 to 13 inclusive}. THINK WRITE Which are the odd numbers starting at 3 and finishing at 13? A = {3, 5, 7, 9, 11, 13}

1WORKEDExample

What is the cardinal number of the set B = {months of the year ending in ER or beginning with J}? THINK WRITE

Which are the required months? January, June, July, September, October, November, December

Count how many there are. n(B) = 7

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2WORKEDExample

2 M a t h e m a t i c a l M e t h o d s U n i t s 1 a n d 2

Equality and equivalence of sets Two sets A, B are said to be equal if they contain the same elements. We write A = B.

Two sets A, B are said to be equivalent if they have the same cardinal number. We write A ↔ B. For example, if A = {2, 5}, B = {5, 2}, C = {3, 4} then A = B, A ↔ C and B ↔ C.

Universal set A universal set is a set which contains all elements under consideration.

The symbol ε denotes the universal set. For example, the universal set could be ε = {whole numbers} if we are dealing with

A = {even numbers} or we could have ε = {sports that involve a ball} if we are con- cerned with the sets A = {tennis, cricket, baseball} and B = {soccer, basketball}.

Complement of a set The complement A′ of a set A is the set of all elements of ε not contained in A. This means that A ∪ A′ = ε.

For example, if ε = {1, 2, 3, 4, 5, 6} and A = {1, 2, 3} then A′ = {4, 5, 6}.

Intersection of sets The intersection of two sets A, B is the set of elements common to A and B and is denoted by A ∩ B.

For example, if A = {1, 2, 3, 4} and B = {3, 4, 5, 6} then A ∩ B = {3, 4}.

Union of sets The union of two sets A, B is the set that contains all elements belonging to A or B or to both A and B. The union is denoted by A ∪ B. Note that an element in the union is listed only once.

The number of elements in A ∪ B does not necessarily equal n(A) + n(B). If there are elements common to A and B, these would be counted twice when finding n(A) + n(B).

To rectify this, we subtract the number of common elements once, that is: n(A ∪ B) = n(A) + n(B) − n(A ∩ B)

For example, if A = {1, 2, 3, 4} and B = {3, 4, 5, 6} then A ∪ B = {1, 2, 3, 4, 5, 6} and n(A ∪ B) = 4 + 4 − 2 = 6.

Disjoint sets Sets A and B are called disjoint sets if A ∩ B = φ; that is, if they have no elements in common.

The sets A = {1, 2} and B = {3, 4} are disjoint sets. For disjoint sets, n(A ∪ B) = n(A) + n(B).

Subsets If every element of set A is an element of set B, then A is said to be a subset of B and is denoted by A ⊂ B. Alternatively, we can say that set B contains set A and write it as B ⊃ A. As an example, if A = {a, b, c} and B = {a, b, c, d, e} then A ⊂ B and B ⊃ A.

C h a p t e r 1 0 E x t e n s i o n — S e t s a n d V e n n d i a g r a m s 3

If ε = {two digit numbers between 10 and 20}, A = {numbers whose digits add up to 5} and B = {even numbers}, list the elements of: a A b A′ c B d B′ e A ∪ B f A ∩ B g (A ∩ B).′

THINK WRITE

a The only number between 10 and 20 whose digits add up to 5 is 14.

a A = {14}

b List all elements in ε that are not in A. b A′ = {11, 12, 13, 15, 16, 17, 18, 19}

c Elements in set B are the even numbers between 10 and 20.

c B = {12, 14, 16, 18}

d List all elements in ε that are not in B; that is, the odd numbers between 10 and 20.

d B′ = {11, 13, 15, 17, 19}

e List all elements in A or in B or in both. Each element is listed once only. This is all elements in set B.

e A ∪ B = {12, 14, 16, 18}

f The only number in both A and B is 14. f A ∩ B = {14}

g List all elements in ε that are not in A ∩ B.

g (A ∩ B)′ = {11, 12, 13, 15, 16, 17, 18, 19}

3WORKEDExample

If ε = {letters in the sentence THE ROAD IS FUN}, A = {letters in the word THREADS}, B = {letters in the word ITS}, C = {letters in the word FASTER}, list: a (A ∪ B) ∩ C b (A ∪ B ∪ C)′ c (A ∩ B ∩ C).′

THINK WRITE

a Determine A ∪ B. a A ∪ B = {T, H, R, E, A, D, S, I} Decide which elements are common to (A ∪ B) and C.

(A ∪ B) ∩ C = {T, H, R, E, A, D, S, I} ∩ {F, A, S, T, E, R} = {A, E, S, T, R}

b Determine A ∪ B ∪ C. b A ∪ B ∪ C = {T, H, R, E, A, D, S, I, F} Find which elements of ε do not belong in A ∪ B ∪ C. Letters in ε that are not in A ∪ B ∪ C are U, O and N.

(A ∪ B ∪ C)′ = {U, O, N}

c Note which elements (T and S) are common to sets A, B and C.

c A ∩ B ∩ C = {T, S}

Find which elements of ε are not in A ∩ B ∩ C.

so (A ∩ B ∩ C)′ = {H, E, R, O, A, D, I, F, U, N}

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1 2

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4WORKEDExample

4 M a t h e m a t i c a l M e t h o d s U n i t s 1 a n d 2

Venn–Euler diagrams The properties of sets can be illustrated by means of a Venn–Euler diagram (commonly referred to as a Venn diagram). It consists of a rectangle representing the universal set ε with all other sets usually represented by circles within the rectangle. The intersection of sets is the common area of overlapping circles.

Some of the main properties of sets are illustrated by means of the Venn diagrams given below.

Universal set

ε ε

εA ⊃

A ε

A A'

Complement A' ε

A

Cardinal number n(A) = 5

* * ***

ε A

3 ∈ A, 8 ∈ A', 5 ∉ A,

1 2 3

5 8

ε A B

A ∩ B

ε A B

A ∪ B

ε B

A

A B⊃

ε A B

(A ∪ B)'

A ∩ B' A ∩ B A' ∩ B

List the elements of: a the universal set ε b A c A′ d B e A ∪ B f A ∩ B g (A ∩ B)′ h A′ ∩ B. THINK WRITE

a All elements in the rectangle are in the universal set, ε.

a ε = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

b All numbers in the circle labelled A are in set A. b A = {1, 2, 3, 4, 5}

c A′ is the region outside the circle and contains the elements 6, 7, 8, 9, 10.

c A′ = {6, 7, 8, 9, 10}

d All numbers in the circle labelled B are in set B. d B = {4, 6, 7, 8}

e All elements in A or B or both are in the union A ∪ B.

e A ∪ B = {1, 2, 3, 4, 5, 6, 7, 8}

f 4 is in both set A and set B. f A ∩ B = {4}

g List all elements not in A ∩ B that are in ε. g (A ∩ B)′ = {1, 2, 3, 5, 6, 7, 8, 9, 10}

h Elements not in A are 6, 7, 8, 9, 10. Elements in B are 4, 6, 7, 8.

h A′ ∩ B = {6, 7, 8}

Numbers that are in both lists are in A′ ∩ B.

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5WORKEDExample ε A B

9

10

2 3

1

6 7 8

5 4

C h a p t e r 1 0 E x t e n s i o n — S e t s a n d V e n n d i a g r a m s 5

ε = {letters of the alphabet from a to k}, A = {a, c, f, g} and B = {a, f, k} List the elements in: a A ∩ B b A ∪ B c (A ∪ B)′ d Draw a Venn diagram for the sets and show that n(A ∪ B) = n(A) + n(B) − n(A ∩ B). THINK WRITE

a Elements in set A and set B are a and f. a A ∩ B = {a, f}

b Elements in A or B or both are a, c, f, g, k. b A ∪ B = {a, c, f, g, k}

c Elements not in A ∪ B that are in ε are b, d, e, h, i, j.

c (A ∪ B)′ = {b, d, e, h, i, j}

d A rectangle will represent ε. d A ∩ B = {a, f} so we require two overlapping circles containing a and f within the rectangle. A = {a, c, f, g} and A ∩ B = {a, f} so the elements c and g lie inside circle A but