#### SET THEORY

##### WHAT IS SET

A set is a well-defined collection of objects.

Here:

• The term "well defined" means that given a set and an object, it is possible to decide whether or not the object belongs to the set.

• The collection of all ENGINEERING students in a particular college.

• The objects constituting a set are called elements or members of the set.

-If x is a member of set A, than we represent as we write x ∈A (read as x belongs to A) and

-If x is not a member of the set A, we write x ∉A (read as x does not belongs to A).

Note:

• Sets are denoted by capital letters A, B, C ,D etc, and

Their members are denoted by small letters a, b, c ,d, etc.

##### REPRESENTATION OF SETS

Set is represented in two forms. These are:

1) Tabular Form or Roster Form.

2) Set-builder Form or Rule Method.

##### 1) TABULAR FORM

Tabulate form of set representation is a method in which , a set is presented by listing all the elements of a set, separating all the elements by commas and enclosing them within curly brackets { }.

For EX

A set of vowels V is represented as: V ={a, e, i, o, u}

##### 2) SET-BUILDER FORM

Set-builder form of set representation is a method in which , a set is presented by one or more variables (says x,y etc)representing an arbitrary member of the set, that is followed by a statement or property which must be satisfied by each element of the set.

FOr EX

A set of vowels V is represented as: V ={x : x ∈a vowel in English}

##### TYPES OF SET

There are many types of set . These are:

##### 1) Finite Set.

A set is said to be finite if it consists of a finite number of elements i.e numbers are countable,

For ex-set of vowels V ={a, e, i, o, u}

##### 2) Infinite Set.

A set is said to be Infinite if Thee set is not a finite set i.e thye numbers of set is not countable

For ex- Number of stars in galaxy.

##### 3) Empty or Null or Void set.

A set having no element is called an empty set or null set or void set. It is denoted by the symbol ∅ or by { }.

For ex- Set of natural numbers between 10 & 11 is a null set.

##### 4) Singleton Set.

A set having one element is called a singleton set.

For ex- {10}is a singleton set.

##### 5) Subset and Super set.

If A and B are two sets such that every element of set A is also an element of set B, Then A is called a subset of B.

Subset is Symbolically represented as A⊆B and it is read as ‘A’ is contained in ‘B’ or A is subset of B.

Superset is Symbolically represented as B⊇A and it is read as ‘B’ is contained in ‘A’ or B is subset of A.

For EXIf A= {1,2,3,5,6} and B{1, 2, 3, 4, 5,6,7} then A is a subset of B.

##### 6) Proper subset.

Set A is called proper subset of a set B if every element of set A is an element of set B whereas every element of B is not an element of A.

Proper set is written as A⊂B and read as ‘A is proper subset of B’

Thus, we can say A is a proper subset of B ⇒ every element of A is an element of B but there is at least one element in B which is not in A.

For EXIf A= {a, b, c, d} and B ={a, b, c, d, e} then A is a proper subset of B and consequently, B is proper superset of A.

##### 7) Power set.

A power set of any set A is the set of all subsets of A and is denoted by P(A).

Power set of set A may be represented as P(A) ={X : X⊆A}

Note-In power set If set A has n elements then P(A) has 2^n elements.

For EXIf A ={1, 2} then its subsets are ∅, {1}, {2}

P(A) = {∅,{1}, {2}, {1,2} which has 2^2 = 4 elements.

##### 8) Multiple Set.

Multiple set may be defined as an unordered collection of elements of a set, in which the multiplicity of an element may be one or more that one or zero.

For EXA = {1, 1, 1, 2, 2, 3 , 3, 4, 4, 4}

Multiplicity of element 1 =3

Multiplicity of element 2 =2

Multiplicity of element 3 =2

Multiplicity of element 4 =3

##### 9) Equal / Equivalent Sets

Two sets A and B are said to be equivalent sets if the number of elements in set A is equal to the no. of elements in set B.It is denoted as A ≈B which is read as ‘A is equivalent to B’.

For EX - A= {a, e, i, o, u} and B= {1, 2, 3, 4, 5} are equivalent sets as both the sets contain 5 elements

##### 10) Equal Sets

Two sets A and B are said to be equal if every element of A is an element of B and every element of B is an element of A.

For EX-A = {2, 4, 6, 8} b= {x : x is even no. < 10} then A=B.

##### 11) Universal Set

When all the sets under consideration are subjects of a larger set then the larger set is called the universal set.

The universal set is denoted by U.

Universal set is super set of all sets.

##### 12) Disjoints Sets

Two sets A and B are called disjoint sets if there are no common element in set A and set B.

For EX-If A = {1, 2, 3, 4} and B ={ 5, 6, 7, 8}. Here set A and B are disjoint sets because there are no common element between the sets A and B.

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