1. ITD1111 Discrete Mathematics & Statistics STDTLP
Unit 1 Sets and Numbers
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ITD1111 Discrete Mathematics & Statistics STDTLP

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1. Sets and Numbers
1.1 Introduction
Sets are used to group objects together. Often the objects in a set have similar properties.
The set is a fundamental discrete structure on which all other discrete structures are built, such as graphs, combinations etc.
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Discrete structures are used in modelling and problem solving. There are many applications on the use of discrete structures in data storage, data communication and manipulation of data.
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1.2 Sets, subsets and elements
Sets
A set is a collection of objects called “elements” or “members”.
We usually use:
uppercase letters A, B, C, , to denote sets, and
lowercase letters a, b, c, , to denote the elements of a set.
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Example
A = {a, e, i, o, u} is a set and the list of all its elements is given.
Example
B = {x : x is an integer, x > 0}
Consider the set C = {1, 3, 5, 7, 9}. We write
3  C to mean that 3 belongs to the set C, and
–5  C to mean that -5 does not belong to C.
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Notation
There are two ways to specify a particular set :
1) To list all its elements if it is possible.
2) To state the property which characterises the elements in the set.
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Equality of sets
Two sets A and B are equal if and only if they have exactly the same elements i.e. if every element belonging to A also belongs to B and if every element belonging to B also belongs to A.
Example
Let A = {a, e, i, o, u}, B = {u, o, i, e, a} and
C = {a, a, e, i, i, o, u}
then A = B = C
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Universal set
A universal set U is a set that contains all the objects under consideration.
Example
U = the set of all students on this campus
A = the set all full-time students on this campus
B = the set of all students in this lecture theatre
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Empty set
A set which contains no element is called the empty set or null set and is denoted by .
Example
Let X = { y : , y is odd }
then X is the empty set and we write
X = .
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Subsets
The set A is said to be a subset of B if every element of A is also an element of B. In symbols,
A  B which is read as “A is a subset of B” or
B  A which is read as “B contains A”.
If A is not a subset of B then it is denoted by A  B.
Example
Let A = {1, 3, 5} and B = {1, 2, 3, 4, 5}
then A  B and B  A
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Disjoint sets
If sets A and B have no elements in common, then we say that A and B are disjoint i.e. no element of A is in B and no element of B is in A.
Example
Let A = {1, 3, 5, 7} and B = {2, 4, 6, 7} then A and B are not disjoint because 7 is in both sets:
7  A and 7  B
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Sets of numbers
Some sets of numbers occur often and special symbols are used for them.
Z = the set of all integers
For example, a subset of Z could be
{ 0, -2, 5, 3, -6, 12, …… }
Z + = the set of positive integers (or natural nos.)
= {x : x  Z and x > 0}
For example, a subset of Z + could be
{ 1, 3, 5, 7, 9, …… }
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N = set of non-negative integers
= {x : x  Z and x  0}
= { 0, 1, 2, 3, 4, …… }
Q = set of rational numbers
= { : x, y  Z and y  0}
e.g. { 1/3, 7/4, –5/9, …… } is a subset of Q
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R = set of real numbers
e.g. { -2/3, -4, 0, 5, 4/9, …… } is a subset of R
The above sets are related as follows:
Z +  N  Z  Q  R
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1.3 Venn diagrams
A Venn diagram is a pictorial representation of sets by sets of points in the plane. In the diagram shown in the following page, the universal set U is represented by the interior of a rectangle, and the other sets are represented by disks lying inside the rectangle.
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Figure 1 represents A  B and A  B.
Figure 2 shows that A and B are disjoint.
Figure 3 shows that A and B are not disjoint. U A B U B A U A B
Figure 1
Figure 2
Figure 3
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1.4 Set operations
Union 
The union of two sets A and B, denoted by A  B, is the set of all elements which belong to A or B or both i.e.
A  B = { x : x  A or x  B }
Figure 4 A  B is shaded.
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Example
Let A = { a, b, c }, B = { 1, 2, 3 } and
C = { a, c, e, 1, 3, 5 }.
then A  B = { a, b, c, 1, 2, 3 }
B  C = { 1, 2, 3, a, c, e, 5 }
C  A = { a, c, e, 1, 3, 5, b }
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Intersection 
The intersection of two sets A and B, denoted by
A  B, is the set of elements common to both
A and B, i.e.
A  B = { x : x  A and x  B } A B U
Figure A  B is shaded.
Note : If A  B =  then A and B are disjoint.
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Example
Let A = { a, b, c, d, e }, B = { c, d, e, f, g } and C = { a, e, i, o, u }.
then A  B = { c, d, e }
B  C = { e }
C  A = { a, e }
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Difference \
The difference of set A and B, denoted by A \ B is the set of elements which belong to A but not to B, i.e.
A \ B = { x : x  A , x  B }.
A \ B is also called the
complement of B with
respect to A.
Figure 6 A \ B is shaded.
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Example
Let S = { a, b, c, d } and T = { c, d, e, f },
then S \ T = { a, b }
T \ S = { e, f }
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Complement
The complement of a set A, denoted by , is the set of elements which do not belong to A. It is the same as the difference of the universal set U and A i.e.
Figure is shaded.
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Example
Let the universal set U be the set containing letters of the English alphabet and A = { a, b, c, x, y, z }.
then = { d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t, u, v, w }
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Example
Use Venn diagrams to represent the following set expressions.
(a) B C A A C B B C A
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(b) A B C
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1.5 Algebra of sets
Sets, under the operations of union, intersection and complement, satisfy the laws listed in the following table. We can use these laws of algebra of sets to simplify complicated set expressions. The following identities can be verified by drawing Venn diagrams.
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ITD1111 Discrete Mathematics & Statistics STDTLP

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Set identities
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Example
Let A, B and C be sets. Use laws of algebra of sets to simplify the following set expressions.
(a)
Commutative law
Distributive law
Note: only two operations are needed.
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(b)
Commutative law
Distributive law
Inverse law
Identity law
Absorption law
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ITD1111 Discrete Mathematics & Statistics STDTLP

31. 1.6 Application of Venn diagrams in counting
Example
In a class of 50 college students, 30 study Pascal, 25 study C and 10 study both computer languages. How many students do not study computer language ?
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No. of students who do not study computer language is
50 – 20 – 10 – 15 = 5 students
U=50
Pascal C 10 20 15
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Example
In a survey of 160 passengers, an airline found that 48 preferred wine with their meals, 78 preferred mixed drinks, and 66 preferred ice tea. In addition, 12 enjoyed wine and mixed drinks, 18 enjoyed mixed drinks and ice tea, and 16 enjoyed ice tea and wine, and 4 passengers enjoyed them all.
a) How many passengers want only iced tea with their meals?
b) How many passengers do not like any of them?
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a) No. of passengers = 36
No. of passengers
= 160 – 24 – 52 – 36 – 12 – 8 – 14 – 4
= 10
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ITD1111 Discrete Mathematics & Statistics STDTLP