Why this chapter matters Sets are collections of objects. Set notation gives us a way of seeing the logical connection between sets. The mathematics of sets is very useful in designing computer circuits and electronic components.
You have probably heard of Alice in Wonderland. Did you know that the author, Lewis Carroll, was actually a lecturer in mathematics at the University of Oxford, England, in the nineteenth century? His real name was Charles Dodgson. He also wrote a mathematics book called Symbolic Logic. Here is a problem from it. 1. All humming birds are richly coloured. 2. No large birds can live on honey. 3. Birds that do not live on honey are dull in colour. What conclusion follows?
Alice in Wonderland Symbolic Logic was about how to write sentences in symbols so that conclusions could be seen more easily. You can use set notation and Venn diagrams to do this and you will learn about them in this chapter. Venn diagrams were invented by the logician John Venn and may be the only mathematical invention to be celebrated in a stained glass window!
Symbolic logic has a very modern application in the design of computer circuits and the construction of electronic components. When engineers talk about NAND and NOR gates they are using ideas that were first developed in the nineteenth century for very different purposes.
A Venn diagram in glass
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Chapter
Ordering and set notation Topics
Level
1 Inequalities
Key words
CORE
equals, greater than, less than
2 Sets
EXTENDED
set, element, subset, empty set, proper subset
3 Venn diagrams
EXTENDED
universal set, Venn diagram, complement, intersection, union
In this chapter you will learn how to: CORE
EXTENDED
Order quantities by magnitude and demonstrate familiarity with the symbols:
Use language, notation and Venn diagrams to describe sets and represent relationships between sets:
=, , , , , . (C1.6 and E1.6)
â—?
Definition of sets, e.g. A = {x: x is a natural number} B = {(x, y): y = mx + c} C = {x: a x b} D = {a, b, c, ‌}. (E1.2)
â—?
Notation Number of elements in set A â€œâ€Śis an element of‌â€? â€œâ€Śis not an element of‌â€? Complement of set A The empty set Universal set A is a subset of B A is a proper subset of B A is not a subset of B A is not a proper subset of B Union of A and B Intersection of A and B
n(A) A’
A A A A A A
B B B B B B. (E1.2) 85
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6.1
Inequalities
You need to know the meaning of these symbols: =
equals
does not equal
is greater than
is less than
is equal to or greater than
is equal to or less than.
The arrow tips in or signs point towards the smaller numbers
EXAMPLE 1
n is a positive integer and n 6. What are the possible values of n? Because n is positive the possible values are 1, 2, 3, 4 and 5. 6 is not included because the sign means ‘less than’. If the question said n 6 then 6 would be included. Sometimes two inequalities are used together. EXAMPLE 2
Find the possible values of x if x is an integer and –3 x 3. This means that x is a whole number between – 3 and 3. The possible values of x are –2, –1, 0, 1, 2, 3. Note that –3 is not in the list but 3 is included because 3 means x can be ‘less than or equal to’ 3.
CORE
EXERCISE 6A Here are three symbols: = . Put the correct symbol between the numbers in each pair. a
3.5 ‌ 3.15
b
180 cm ‌ 2m
c
5×7‌6×6
d
5km ‌ 5000m
e
1 3
of 27 ‌ 34 of 12
f
42 ‌ 8
g
10 × 10 ‌ 8 × 12
h
ďŁĽâˆš6  4 ‌ 32
1 3 1 Here are three fractions: , , . 3 5 2 Use them to fill in the gaps below to list them in order, smallest first. ‌ ‌ ‌ 86
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CHAPTER 6: Ordering and set notation
CORE
d is the number scored when a normal six-sided dice is thrown. List the possible values of d in each case. a
d 4
b
d 3
c
d 5
d
d 5
e
2 d 4
f
3 d 6
g
1 d 4
h
5 d 6
The table shows whether babies of a particular age are underweight or overweight. Mass 6.5 kg 6.5 kg mass 8.5 kg Mass 8.5 kg
Underweight Normal Overweight
Are babies of these masses underweight, overweight or normal? a
6.3 kg
b
9.3 kg
c
7.8 kg
d
8.5 kg
e is an even number and 20 e 30 and e 24. List the possible values of e. In a bag of balls each ball has a number n where 1 n 49. a
What is the largest number on a ball?
b
The number on the first ball is a multiple of 5 and n 40. What are the possible values of n?
c
The number on the second ball is a multiple of 3 and n 10. What are the possible values of n?
d
For the third ball 15 n 20. What are the possible values of n?
True or false? State which in each case. a
3 –3
b
–3 –5
c
1.99 2
d
2 ďŁĽâˆš5 3
e
202 300
f
200 minutes 3 hours
List all the possible values for an integer x in the following cases. a
5 x 9
b
26 x 28
c
–8 x –4
d
–2 x 2
e
17 x 18
f
32.5 x 33.5
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6.2
Sets
A set is any collection of items. They could be numbers, objects or symbols. The items in a set are called elements. The elements in a set can be listed or described inside brackets like this { }. The elements are separated by commas, e.g. {vowels in English} = {a, e, i, o, u}. You will usually use capital letters to stand for sets, e.g. N = {1, 3, 5, 7, 9}. You should use dots to show when you cannot list all the elements, e.g. {even numbers} = {2, 4, 6,‌}. You can describe sets using inequality symbols e.g. {integer x : 5 x 10} = {5, 6, 7, 8, 9, 10} can be read as ‘the set of integers between 5 and 10 inclusive’.
Describing elements Suppose A = { 2, 4, 6, 8, 10} and B = {1, 2, 3, 4, 5, 6}. 8 is an element of A but not of B. This can be expressed as: 8 A (8 is an element of A) 8 B (8 is not an element of B) The number of elements in a set can be shown as n(set), e.g. the number of elements in set X is n(X). So for sets A and B above, n(A) = 5 and n(B) = 6. Suppose C = {6, 8, 10}. All the elements of C are also elements of A. Then you can say that C is a subset of A and write this as:
C A.
is a bit like the sign we use with numbers.
Suppose P = {red, blue, yellow}. P has eight subsets: {red, blue}, {red, yellow} and {blue, yellow} have two elements. {red}, {blue}, {yellow} have one element. You also include the empty set, , which has no elements and P itself is the final subset. All the subsets except P are called proper subsets and the symbol to show a proper subset is . So {red} P ({red} is a proper subset of P)
is a bit like the sign we use with numbers.
but P P (P is not a proper subset of P)
EXTENDED
EXERCISE 6B Describe these sets in words. a
{Monday, Tuesday, Wednesday...}
b
{1, 3, 5, 7...}
c
{Mercury, Venus, Earth, Mars...}
d
{North, South, East, West}
e
{1, 2, 3, 4, 5, 6}
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CHAPTER 6: Ordering and set notation
a
A = {integer x: 5 x 10}
b
B = {factors of 12}
c
C = {prime numbers between 20 and 30}
d
D = { x: x2 = 9}
Using the sets in question 2, find n(A), n(B), n(C) and n(D).
EXTENDED
List the elements of each of these sets.
M = {months of the year} a
Write down n(M).
b
List a subset of M with three elements.
X = {6, 7, 8, 9} a
List all the subsets of X with three elements.
b
List all the subsets of X with one element.
Y = {left, right} a
List the four subsets of Y.
b
Which subset is not a proper subset?
A set has six elements. a
How many subsets does it have with one element?
b
How many subsets does it have with five elements?
Here are five sets.
A = {a, b, d, g, l, m} B = {d, l, m, p, r} C = {a, m, p} D = {a, d, l, m} E = {d, l, p} a
From the list identify: i
a proper subset of A
ii
a proper subset of B.
b
X is one of the sets shown and n(X) = 5. Which set is X?
c
stands for an element with these properties: A
B
C
D
E
What letter does stand for?
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6.3
Venn diagrams
The set that contains all the elements of a number of subsets is called a universal set, . Take the universal set of positive single digit integers:
= {1, 2, 3, 4, 5, 6, 7, 8, 9} This includes several subsets including:
A = {x: x is even} B = {x: x 5) You can show these sets in a Venn diagram.
A 6
2
8
4 7
B
1
A
B
3 5 9
C
Each of the universal set is in A, in B, in both, or in neither. element A B A’ is the complement of set A and has the elements in that are not in A: 2 8 A’ = {1, 3, 5, 7, 9}
1
4
P
3 6 A B is the intersection 5 of A and B and contains elements that are in A and B: A B = {2, 4} 7
E
S
A B is the9union of A and C B and contains elements that are in A or B or both: A B = {1, 2, 3, 4, 5, 6, 8} (A of A B and contains elements B)’Ais the complement B that are not A in A and B 1together: B 6
2
8
4
(A B)’ = {7, 9}
3
A B
5
You can also use Venn diagrams with three C sets. 7
A
means i tersection means nion A
9
C
A 2
8
4
6
A 9g
i
C
B
1 3
s 7 a
n
C
B
Suppose you add C = { x: 3 x (A7} to theC example. B)
B
P
B
5 p r
E
S
A
B
A
C
C e o
C = {3, 4, 5, 6, 7} A BA = {1, 2, 3, 4, 5, 6, B8} C B (A B) C = {3, 4, 5, 6}
A B C
90
(A B) C
B
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B
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2 7
8
A
8
9
4
6
3
9 1
7
CHAPTER 6: Ordering and set notation
1
C
B 4 6 by3shading You can illustrate6this different 51 2 3 7 8 4 5B A 9 C 7
9
B 2
8
4
6
A
EXERCISE 6C n
9
g
A A
(A B) C
s
g
a i
a
eB
A 6
A A
B C B
7
B
B
vii A’ B pB
10 r a ii e
B
iii
A B
viii
A B’.
iv
A B
v
n(A B)
iii
A
A’
B
n(B’).
o name =A {letters i in the of a city state}. What is the city state? 8 B
C
6 2 5 11 12
3 11
A
94 1
10
C
5
B
7
8 9
BC
= {f, r, a, c, t, i, o, n, s} A {r, a, 6t, i, o} C =A 2 D = {f, i, r, s, 4t}
B B 10
12
a
1 Copy the diagram 3 7 and put the elements in the correct places.
b
List the elements of: 5 i
S
B
(A B) C
ii
g i 12 n(A) 1 3
c
C
C
s n 2 Write down: 4
C
o
C
A
A
A B
r
vi A (A B)’
c
E
A P B
B B
p
List the elements of: i
b
S
BC
o n
B
CC
e
i
C
C
r
a
A B
A B
5
E
A P
EXTENDED
3
(A B) C
1
C
p
s
regions of the diagram.
C
7
A B
A
S
C
B
2
A
E
P
5
11
(C D)’
9
8
C
ii
C’
ii
n((C D)’).
iii
C’ D.
Find: i
n(C D) A
B
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A n
s
g
a
9
C
r
A
B
e
CHAPTER 6: Ordering and set notation i
7
B
p
o
A
B
EXTENDED
= {integer x: 1 x 12} C A = {multiples of 2} B = {multiplies of 3}
A B
B
C
a
Show these sets on a Venn diagram.
b
Describe set A B.
A
4
12 3 11
a
1 5
C
B
6
2
(A B) C
A
10 7
s a
8 9
i
C
C
A
r e o
B
List the elements of each of these sets. A
i
B
A
ii
B
iii
A B
iv
A C
v
B C
vi
C’
vii
A B C
viii
(A B) C
ix
(B C)’
x
(B C)’ A
2
4 3
= {integer x: ... x ‌}
i ii
A = {factors of ...}
iii
C = {...numbers}
11
On copies of this Venn diagram, shade each region. a
A B
b
A’ B
c
(A B)’
d
(A B)’
e
A’ B’
f
A’ B’
B
6
12
Complete these descriptions.
b
n g
B
p
1 5
10 7 8 9
C
A
B
A
B
Which of your answers in question 5 are identical?
On copies of this Venn diagram, shade these regions. A
a c e
6 8
(A B) 1 C
B
2
3 (A C) (A 4 B)
b
A’ (B C)
d
(A B C)’
5
(B C)’ A 7
9
A
C
B 2
8
92
6
4
1 3
5
P
E
S
7 50372_P084_093.indd 92
9
C
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8
4
A 7
A
6
2
8
4
A
1
A
7 19
B
p
s
n
CHAPTER 6: Ordering and set notation
r In4 this Venng diagram: a 5
P
e
C
Explain why n(P E) = 1. 7 B
9 b
C P S = . Explain why
c
Describe the elements of E’ S.
A B
C
A
10
A B
a
A n
c s
g
ea
9
(A B8) Cb
C
e
A
p
s
S
C
B
A
B A
d
A B=B
f
B C=
C B
A
B
r
a i
B
A
B
B
e
o
n(A) = 50
C
5
B C=
g
B
7
r
A i o
C
3
11 p B=B C A
n
E
P
B
a
B
S
B d Write down aBnumber x such that x P and x E and x S. (A B) C A B 6 2 Look atCthis Venn 4 diagram.A B
A
A E
= {positive integers} B
12 Say whether each is true or false. C 1 of these statements
B C
5
A 3
A
3
B
i o P7= {prime numbers} 2 C 1 E8 = {even numbers} 4 S = 6{square C 3 5 numbers}.
9
C
B
EXTENDED
6
9
2
8
(A B ) C
5
B n(B) = 40
n(A B) = 27
What is n(A B)?
A 2
6
12 3
A
B
A and B are two sets. Write, as simply as possible: 4 a 1
10
A A’
b
B B’
c
((A U B)’)’
d
(A B) B
e
(A B) A.
7
= {x: x is a8 positive integer} B
5 6 112 E = {9x: x is a multiple of 2} 4C 10 12T = {x: x is a multiple of 3} F =3{x:1x is7 a multiple of 5} a
A
b
8
5 a Venn diagram and place these numbers in it: 10, 11, 12, 13, 14, 15. Draw
11
9B
CompleteCthis description.
E T = {x: x is a multiple of …..}
A
c
Write a description of T F.
d
Find an element of E T F.
e
Write a description of E T F.
B
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