SET THEORY

In general, set are just collection or group of objects.

Set are fundamental discrete structure on which most of the discrete structures are built
Definitely set are those fundamental structure through which you can perform any analysis of graph
theory, probability, functions or relations.

Definition: A well defined, unordered collection of distinct objects or elements.

 The object in a set are called its members or elements.

Some Example of Sets

 A set of all positive integers.

 A set of all the planets in the solar system.
 A set of all the states in India.
 A set of all the lowercase letters of the alphabet.

The following points are noted while writing a set:

 Sets are usually denoted by capital letters A, B, S, etc.
 The elements of a set are usually denoted by small letters a, b, t, u, etc.
 Elements are written in curly brackets separated by comma.
 Examples: A = {a, b, d, 2, 4}
 B = {math, religion, literature, computer science }The following points are noted while writing a
set.

Set Representation
There are two ways to represent a set.
1. Roster or tabular form:
The set is represented by listing all the elements comprising it. The elements are enclosed within
braces and separated by commas.
Example 1 − Set of vowels in English alphabet, A={a,e,i,o,u}
Example 2 − Set of odd numbers less than 10, B={1,3,5,7,9}
2. Set Builder Representation:
The set is defined by specifying a property that elements of the set have in common. The set is
described as A={x:p(x)}
Example 1 − The set {a,e,i,o,u} is written as −A={x:x is a vowel in English alphabet}
Example 2 − The set {1,3,5,7,9} is written as −B={x:1≤x<10 and (x%2)≠0}
If an element x is a member of any set S, it is denoted by x∈S and if an element y is not a
member of set S, it is denoted by y∉S.

Some Important Sets

 N − the set of all natural numbers = {1,2,3,4,.....}
Z − the set of all integers = {.....,−3,−2,−1,0,1,2,3,.....}
Z+ − the set of all positive integers

Q − the set of all rational numbers

R − the set of all real numbers

W − the set of all whole numbers.

Cardinality of a Set
It is the number of elements of the set. The number is also referred as the cardinal number. Cardinality
of a set S, denoted by |S|or n(S), If a set has an infinite number of elements, its cardinality is ∞.
Example − |{1,4,3,5}|=4,|{1,2,3,4,5,…}|=∞

Types of set
Sets can be classified into many types. Some of which are finite, infinite, subset, universal, proper,
singleton set, etc.
Finite Set
A set which contains a definite number of elements is called a finite set.
Example − S={x|x∈N and 70>x>50}
Infinite Set
A set which contains infinite number of elements is called an infinite set.
Example − S={x|x∈N and x>10}

An empty set contains no elements. It is denoted by ∅. As the number of elements in an empty set is
Empty Set or Null Set

finite, empty set is a finite set. The cardinality of empty set or null set is zero.
Example − S={x|x∈N and 7<x<8}=∅
Singleton Set or Unit Set
Singleton set or unit set contains only one element. A singleton set is denoted by {s}.
Example − S={x|x∈N, 7<x<9} = {8}
Equal Set
If two sets contain the same elements they are said to be equal.
Example − If A={1,2,6} and B={6,1,2}, they are equal as every element of set A is an element of set B and
every element of set B is an element of set A.

Equivalent Set
If the cardinalities of two sets are same, they are called equivalent sets.
Example − If A={1,2,6} and B={16,17,22}, they are equivalent as cardinality of A is equal to the cardinality
of B. i.e. |A|=|B|=3
Subset
A set X is a subset of set Y (Written as X⊆Y) if every element of X is an element of set Y.
Example 1 − Let, X={1,2,3,4,5,6} and Y={1,2}. Here set Y is a subset of set X as all the elements of set Y is
in set X. Hence, we can write Y⊆X.

Example 2 − Let, X={1,2,3} and Y={1,2,3}. Here set Y is a subset (Not a proper subset) of set X as all the
elements of set Y is in set X. Hence, we can write Y⊆X.

Proper Subset
The term “proper subset” can be defined as “subset of but not equal to”. A Set X is a proper subset of
set Y (Written as X⊂Y) if every element of X is an element of set Y and |X|<|Y|.
Example − Let, X={1,2,3,4,5,6} and Y={1,2}. Here set Y⊂X since all elements in Y are contained in X too
and X has at least one element is more than set Y.

Universal Set
It is a collection of all elements in a particular context or application. All the sets in that context or
application are essentially subsets of this universal set. Universal sets are represented as U.
Example − We may define U as the set of all animals on earth. In this case, set of all mammals is a subset
of U, set of all fishes is a subset of U, set of all insects is a subset of U, and so on.
DISJOINT SET
Two sets A and B are called disjoint sets if they do not have even one element in common.
Example − Let, A= {1, 2, 6} and B= {7, 9, 14}, there is not a single common element.

POWER SET
The set of all subset of a given set A is called power set of A and denoted by P(A).
If n(A)= m then n[P(A)]= 2m.

all possible subset of A are ∅, {a}, {b}, {a,b}.

eg. Write down all possible subsets of A={a,b} .

OPERATION OF SETS
 UNION
 INTERSECTION
 SET DIFFERENCE
 COMPLEMENT
UNION OF SETS
The union of two sets A and B is the set of all those elements which are either in A or in B or in both A

A∪B= {x:x∈A or x∈B}

and B. it is denoted by C.

Eg. A={2,3,5,7} and B={1,2,4,8} then A∪B= {1,2,3,4,5,7,8}.

INTERSECTION OF SETS

A ∩B= {x:x∈A and x∈B}

The intersection of two sets A and B is the set of all those elements which are common to both A and B.

Eg. A={2,3,5,7} and B={1,2,4,8} then A∩B= {2}.

DIFFERENCE OF SETS
The set difference of sets A and B (denoted by A–B) is the set of elements which are only in A but not in
B. Hence, A−B={x|x∈A AND x∉B}
Example − If A={10,11,12,13} and B={13,14,15}, then (A−B)={10,11,12} and (B−A)={14,15}. Here, we can
see (A−B)≠(B−A).

Let A and B be two sets. Then (A-B) ∪ (B-A) is called symmetric difference of A and B. it is
SYMMETRIC DIFFERENCE OF SETS

denoted by A∆B. i.e.

A∆B= (A∪B)-(A ∩B)

COMPLEMENT OF A SET
The complement of a set A (denoted by A′) is the set of elements which are not in set A. Hence, A ′={x|
x∉A}.

More specifically, A′=(U−A) where U is a universal set which contains all objects.
Eg. U={1,2,3,4,5,6,7,8,} and A={2,4,6,8} then A’={1,3,5,7}.

ALGEBRA OF SET THEORY

COMMUTATIVE: let A and B are two sets.
I. A∪B= B∪A

Proof: let x∈ A∪B → x∈A or x∈B

II. A ∩B= B∩A

→ x∈B or x∈A
→x∈ B∪A
A∪B⊆ B∪A……………………………..1
Similarly we have B∪A⊆ A∪B………………………………2
From 1 and 2
A∪B= B∪A
ASSOCIATIVITY

I. (A∪B) ∪C = A∪(B ∪C)

II. (A ∩B) ∩C= A ∩(B ∩C)
IDEMPOTENT
I. A∪A=A
II. A∩A=A
IDENTITY LAWS
I. A∪∅=A
II. A∩U=A

I.
DEMORGAN’S LAWS
(A∪B)’= A’∩B’
 II. (A ∩B)’= A’∪ B’
DISTRIBUTIVE LAWS

A ∩(B∩C)= (A ∩B) ∪(A∩C)

I. A∪(B∩C)= (A∪B) ∩(A∪C)
II.
CARTESIAN PRODUCT

(a,b) where a∈A and b∈B.

If A and B are two non-empty sets, then the Cartesian product of A and B is the set of all ordered pairs

MULTISETS
A collection of object that are not necessarily distinct, is called a Multiset.
Eg. A={1, 1, 2, 3,3,3,4}
Multiplicity: Number of occurrences of an element in the set.
μ (1) =2.

EQUALITY of MULTISETS
If the number of occurrence of each element is the same in both then the multiset are equal.
Eg. A= {a,b,a,a} and B={a,a,b,a}
A=B
MULTISUBSET
A multiset P is said to be a multisubset of Q, if multiplicity of each element in P is less or equal to its
multiplicity in Q.
Eg. P= {1, 2, 2,3} and Q= {1, 1, 1,2,2,3}
P⊆Q
OPERATIONS ON MULTISET
Intersection of multisets
The intersection of two multisets A and B, is a multiset such that the multiplicity of an element is equal
to the minimum of the multiplicity of an element in A and B and is denoted by A ∩ B.
Eg. Let A = {l, l, m, n, p, q, q, r}
B = {l, m, m, p, q, r, r, r, r}
A ∩ B = {l, m, p, q, r}.

element is equal to the maximum of the multiplicity of an element in A and B and is denoted by A ∪ B.
Union of Multisets: The Union of two multisets A and B is a multiset such that the multiplicity of an
Example:
Let A = {l, l, m, m, n, n, n, n}

A ∪ B = {l, l, m, m, m, n, n, n, n}
B = {l, m, m, m, n},

Difference of Multisets: The difference of two multisets A and B, is a multiset such that the multiplicity
of an element is equal to the multiplicity of the element in A minus the multiplicity of the element in B if
the difference is +ve, and is equal to 0 if the difference is 0 or negative

Example:
Let A = {l, m, m, m, n, n, n, p, p, p}
B = {l, m, m, m, n, r, r, r}
A - B = {n, n, p, p, p}

Sum of Multisets: The sum of two multisets A and B, is a multiset such that the multiplicity of an
element is equal to the sum of the multiplicity of an element in A and B

Example:
Let A = {l, m, n, p, r}
B = {l, l, m, n, n, n, p, r, r}
A + B = {l, l, l, m, m, n, n, n, n, p, p, r, r, r}
VENN DIAGRAMS
In order to express the relationship among sets, we represent them pictorially by means of diagrams,
called Venn Diagrams.
 The universal set is represented by a rectangular region.
 Other set is by circles inside the rectangle
le.