Discrete mathematics for diploma students

Discrete
Mathematics
FWD 111
Lecture [05]
Sets

Sets are used to group objects together. Often, but not
always, the objects in a set have similar properties.
For instance, all the students who are currently enrolled
in your school make up a set.
Likewise, all the students currently taking a course in
discrete mathematics at any school make up a set.
In addition, those students enrolled in your school who are
taking a course in discrete mathematics form a set that can be
obtained by taking the elements common to the first two
collections
Introduction

Definition A set is an unordered collection of distinct
objects, called elements or members of the set.
A set is said to contain its elements.
Sets are denoted using uppercase letters.
Elements of sets are denoted using lowercase letters.
a ∈ A
denotes that a is an element of the set A.
a ∉ A
denotes that a is not an element of the set A.
Sets

One way is to list all the members of a set, when this is possible.
For example, the notation {a, b, c, d} represents the set with
the four elements a, b, c, and d.
EX The set V of all vowels in English
V = {a, e, i, o, u}.
EX The set O of odd positive integers less than 10
O = {1, 3, 5, 7, 9}.
EX The set M of positive integers less than 100
M = {1, 2, 3,…, 99}.
Sets description

Another way to describe a set is to use set builder notation.
We characterize all those elements in the set by stating the property or properties
they must have to be members.
The general form of this notation is {x ∣ x has property P} and is read “the set of all
x such that x has property P.”
For instance, the set O of all odd positive integers less than 10 can be written as
O = {x ∣ x is an odd positive integer less than 10},
or, specifying the universe as the set of positive integers, as
O = {x ∈ Z+ ∣ x is odd and x < 10}.
We often use this type of notation to describe sets when it is impossible to list all
the elements of the set.
For instance, the set Q+ of all positive rational numbers can be written as
Q+ = {x ∈ R ∣ x = p/q , for some positive integers p and q}.
Sets description

Definition Two sets are equal if and only if they have the
same elements. Therefore, if A and B are sets, then A and B are
equal if and only if ∀x(x ∈ A ↔ x ∈ B). We write A = B if A and
B are equal sets.
EX The sets {1, 3, 5} and {3, 5, 1} are equal, because they have
the same elements.
Note that the order in which the elements of a set are listed
does not matter.
Note also that it does not matter if an element of a set is listed
more than once, so {1, 3, 3, 3, 5, 5, 5, 5} is the same as the set {1,
3, 5} because they have the same elements.
Sets equality

There is a special set that has no elements.
This set is called the empty set, or null set, and is
denoted by Ø
The empty set can also be denoted by { }.
Often, a set of elements with certain properties turns out
to be the null set.
For instance, the set of all positive integers that are
greater than their squares is the null set.
The empty set

Sets can be represented graphically using Venn diagrams.
In Venn diagrams the universal set U, which contains all the
objects under consideration, is represented by a rectangle.
Inside this rectangle, circles are used to represent sets.
EX Draw a Venn diagram
that represents V, the set of
vowels in the English
alphabet.
Venn Diagrams

Definition The set A is a subset of B, and
B is a superset of A, if and only if every
element of A is also an element of B.
The notation A ⊆ B indicates that A is a
subset of the set B.
Theorem For every set S, (i) Ø ⊆ S and (ii) S ⊆ S.
We have these useful rules for determining whether one set is a
subset of another:
To show that A ⊆ B, show that if x belongs to A then x also belongs to B.
To show that A ⊈ B, find a single x ∈ A such that x ∉ B.
To show that two sets A and B are equal, show that A ⊆ B and B ⊆ A.
Subsets

Definition Let S be a set. If there are exactly n distinct
elements in S where n is a nonnegative integer, we say
that S is a finite set and that n is the cardinality of S.
The cardinality of S is denoted by |S|.
EX
Let A be the set of odd positive integers < 10. Then |A| = 5.
Let S be the set of letters in the English. Then |S| = 26.
The null set has no elements, it follows that |Ø| = 0.
The Size of a Set (Cardinality)

Definition Let A and B be sets. The union of the sets A
and B, denoted by A ∪ B, is the set that contains those
elements that are either in A or in B, or in both.
EX The union of the sets {1, 3, 5} and {1, 2, 3}
is the set {1, 2, 3, 5}
{1, 3, 5} ∪ {1, 2, 3} = {1, 2, 3, 5}.
Union

Definition Let A and B be sets. The intersection of the
sets A and B, denoted by A ∩ B, is the set containing
those elements in both A and B.
EX The intersection of the sets {1, 3, 5} and {1, 2, 3} is the
set {1, 3}
 {1, 3, 5} ∩ {1, 2, 3} = {1, 3}.
Intersection

Definition Let A and B be sets. The difference of A and
B, denoted by A − B, is the set containing those elements
that are in A but not in B.
EX The difference of {1, 3, 5} and {1, 2, 3} is the set {5}
{1, 3, 5} − {1, 2, 3} = {5}.
{1, 2, 3} − {1, 3, 5} = {2}.
Difference

Definition Let U be the universal set. The complement of
the set A, denoted by 𝐀, is the complement of A with
respect to U.
Therefore, the complement of the set A is U − A.
EX Let A be the set of positive integers greater than 10.
𝐀 = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}.
EX Let A = {a, e, i, o, u}.
𝐀 = {b, c, d, f, g, h, j, k, l, m, n, p, q, r, s, t, v, w, x, y, z}.
Complement

1.List the members of these sets.
a) {x ∣ x is a real number such that x2 = 1}
b) {x ∣ x is a positive integer less than 12}
c) {x ∣ x is the square of an integer and x < 100}
d) {x ∣ x is an integer such that x2 = 2}
2.Suppose that A = {2, 4, 6}, B = {2, 6}, C = {4, 6}, and D = {4, 6, 8}.
Determine which of these sets are subsets of which other of
these sets.
3.Determine whether each of these statements is true or false.
a) x ∈ {x} b) {x} ⊆ {x} c) {x} ∈ {x} d) {x} ∈ {{x}} e) Ø ⊆ {x} f) Ø ∈ {x}
4.Let A = {1, 2, 3, 4, 5} and B = {0, 3, 6}. Find
a) A ∪ B. b) A ∩ B. c) A − B. d) B − A.
Exercises

5. Let A = {a, b, c, d, e} and B = {a, b, c, d, e, f, g, h}. Find
a) A ∪ B. b) A ∩ B. c) A − B. d) B − A.
6. Find the sets A and B if A − B = {1, 5, 7, 8}, B − A = {2, 10},
and A ∩ B = {3, 6, 9}.
7. Let A = {0, 2, 4, 6, 8, 10}, B = {0, 1, 2, 3, 4, 5, 6}, and C = {4,
5, 6, 7, 8, 9, 10}. Find
a) A ∩ B ∩ C. b) A ∪ B ∪ C. c) (A ∪ B) ∩ C. d) (A ∩ B) ∪ C.
8. Draw the Venn diagrams for each of these combinations
of the sets A, B, and C.
a) A ∩ (B ∪ C) b) A ∩ B ∩ C c) (A − B) ∪ (A − C) ∪ (B − C)
Exercises

Discrete mathematics for diploma students

Discrete mathematics for diploma students

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Discrete mathematics for diploma students