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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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