This document introduces the concept of sets in discrete mathematics. It defines what a set is and discusses how sets are represented and notated. It also covers basic set operations like union and intersection. Some key points include: - A set is a collection of distinct objects, called elements. - Sets can be represented either by listing elements within curly braces or using set-builder notation describing a common property. - Basic set operations include union, intersection, subset, power set, and the empty set. Union combines elements in sets while intersection finds elements common to sets.

1. DISCRETE MATHMATICS
Introduction to Set Theory

2. HISTORY OF SETS
The theory of sets was
developed by German
mathematician Georg
cantor (1845-1918). He first
encountered sets while
working on “problems on
trigonometric series” . sets
are being used in
mathematics problem since
they were discovered.

3. INTRODUCTION TO SET
THEORY
•A set is a collection of objects.
•The objects in a set are called
elements of the set.
•Set theory deals with
operations between, relations
among, and statements about
sets.

4. SETS REPRESENTATION
There are two ways to represent sets
 Roster or tabular form.
 Set-builder form.

5. s
ROSTER OR TABULAR FORM
In roster form, all the elements of set are
listed, the elements are being separated by
commas and are enclosed within braces { }.
e.g. : set of 1,2,3,4,5,6,7,8,9,10.
{1,2,3,4,5,6,7,8,9,10}

6. SET-BUILDER FORM
In set-builder form, all the elements of a set
possess a single common property which is
not possessed by an element outside the set.
e.g. : set of natural numbers k
k= {x : x is a natural number}

7. BASIC NOTATIONS FOR SETS
• For sets, we’ll use variables S, T,
U, …
• We can denote a set S in writing
by listing all of its elements in
curly braces:

8. – {a, b, c} is the set of
whatever 3 objects are
denoted by a, b, c

9. SETS
Collection of object of a particular
kind, such as, a pack of cards, a
crowed of people, a cricket team
etc. In mathematics of natural
number, prime numbers etc.

10. EXAMPLES OF SETS
IN MATH
• N = The set of natural numbers.
= {1, 2, 3, …}.
• W = The set of whole numbers.
={0, 1, 2, 3, …}
• Z = The set of integers.
= { …, -3, -2, -1, 0, 1, 2, 3, …}
• Q = The set of rational numbers.
={x| x=p/q, where p and q are elements of
Z and
q ≠ 0 }
• H = The set of irrational numbers.
• R = The set of real numbers.
• C = The set of complex numbers.

11. TYPES OF SETS
• Empty sets.
• Finite &Infinite sets.
• Equal sets.
• Subset.
• Power set.
• Universal set.

12. THE EMPTY SET
• The empty set is a special set. It
contains no elements. It is usually
denoted as { } or
.
• The empty set is always considered
a subset of any set.
• Do not be confused by this
question:
• Is this set {0} empty?
• It is not empty! It contains the
element zero.

13. FINITE & INFINITE SETS
A set which is empty or consist of a
definite numbers of elements is called
finite otherwise, the set is called
infinite.
•e.g. : let k be the set of the days of the
week. Then k is finite
• let R be the set of points on a
line. Then R is infinite

14. DEFINITION OF SET
EQUALITY
• Two sets are declared to be equal if and
only if they contain exactly the same
elements.
• In particular, it does not matter how the
set is defined or denoted.
• For example: The set {1, 2, 3, 4} =
{x | x is an integer where x>0 and
x<5 } =
{x | x is a positive integer whose
square
is >0 and <25}

15. SUBSETS
A set R is said to be subset of a set
K if every element of R is also an
element K.
R ⊂ K
This mean all the elements of R
contained in K.

16. POWER SET
The set of all subset of a given set is
called power set of that set.
The collection of all subsets of a set K is
called the power set of denoted by P(K).In
P(K) every element is a set.
If K= [1,2}
P(K) = {ϕ, {1}, {2}, {1,2}}

17. UNIVERSAL SET
Universal set is set which contains all object,
including itself.
e.g. : the set of real number would be the
universal set of all other sets of number.
NOTE : excluding negative root

18. THE UNION OPERATOR
• For sets A, B, their union A U B is the set
containing all elements that are either in A,
or (“U”) in B (or, of course, in both).
17
or (“U”) in B (or, of course, in both).
• Formally, U A,B: A U B = {x | x U A U x U B}.

19. SOME PROPERTIES OF THE
OPERATION OF UNION
1) A U B = B U A ( commutative law )
2) ( A U B ) U C = A U ( B U C ) ( associative law )
3) A U ϕ = A ( law of identity element )
4) A U A = A ( idempotent law )
5) U U A = A ( law of U )

20. UNION EXAMPLES
• {a,b,c} ∩ {2,3} = {a,b,c,2,3}
• {2,3,5} ∩ {3,5,7} = {2,3,5,3,5,7}
={2,3,5,7}

21. s
THE INTERSECTION
OPERATOR
• For sets A, B, their intersection A ∩ B is
the
set containing all elements that are
simultaneously in A and (“∩”) in B.
19
simultaneously in A and (“∩”) in B.
• Formally, ∩ A,B: A ∩ B ∩{x | x ∩ A ∩ x ∩
B}.
• Note that A ∩ B is a subset of A and it is
a
subset of B:
A, B: (A ∩ B ∩ A) ∩ (A ∩ B ∩ B)

22. INTERSECTION EXAMPLES
• {a,b,c} ∩ {2,3} = ___
• {2,4,6} ∩ {3,4,5} = ______

23. SOME PROPERTIES OF THE
OPERATION OF INTERSECTION
1) A ∩ B = B ∩ A ( commutative law )
2) ( A ∩ B ) ∩ C = A ∩ ( B ∩ C ) ( associative law )
3) Φ ∩ A = Φ, U ∩ A = A ( law of Φ and U )
4) A ∩ A = A ( idempotent law )
5) A ∩ ( B U C ) = ( A ∩ B ) U ( A ∩ C ) ( distributive law )