Set Overview

SET 

•A well defined collection of distinct objects is called Set.

•The members are called the elements of Set. The elements of the set are generally denoted by the small letters a,b,c,…..x,y,z.

•The Sets are generally denoted by the Capital letters A,B,C,…..X,Y,Z.

•If x is an element of A, then we say x belongs to A and we write x∈A. If x is not in A, then we say x doesn’t belong to A and we write x∉A.

 EXAMPLES OF SET 

•N= The set of natural numbers = {1,2,3,………}•W= The set of whole numbers = {0,1,2,……….}•ℤ= The set of integers= {………,-2,-1,0,1,2,……..}•ℚ= The set of  rational numbers={ p/q ; p ,q ∈ Z and q≠0}

•ℝ= The set of real numbers ={x: x is either rational or irrational}

ℂ= The set of complex numbers={x+iy ; x,y ϵ ℝ and i=√(-1)}          [ i=imaginary number  called iota.]

REPRESENTATION OF SET

uRoster or Tabular Method

  In this method, a set is represented by listing all its elements in curly brackets and separating them by commas.

For ex: The set containing 3,4,5 and 6 can be written as A={3,4,5,6}

uSet  Builder Method:

In this method, a set is represented by stating all the properties which are satisfied by the elements of the set and not by any other elemets out of the set.

If A={1,2,3,4,5,6} then we write,

A={x:x is a natural number ≤ 6} OR, A={x:x ∈ℕ, x≤ 6}

FINITE AND INFINITE SET

•A set is said to be a finite set if its elements are countable or its elements are finite or its cardinality is fixed.

A={3,5,8}      B={a, b, c, d, ………..}

•A set is said to be an infinite set if its ielements are uncountable or its elements are infinite or its cardinality is  not fixed.

A={1,2,3,………..}

  Other examples: The set of whole numbers, the set of integers etc.

SOME IMPORTANT SETS

■Null/Empty/Zero/Void sets:    A set having no element is called null set. It is denoted by ∅ or { }.   Eg: A={x: 2<x<3 , xϵ ℕ}={ }■Singleton Set:    A set having only one  element is called  singleton set.

  Eg: A={x: 2<x<4 , xϵℕ}={3}   B={0}

■Equivalent Sets:      Two sets A  and B are said to be equivalent sets if n(A)=n(B).

Eg: If A={1,2,3} and B={a,b,c} then  we can write  A~B.

■Equal Sets:    Two sets A  and B are said to be equal sets if they have same elements.

Eg: If A={1,2,3} and B={2,1,3} then we can write A=B. In other words,if A⊆B and B⊆ A then this implies A=B   

■Subsets:   A set ‘A’ is said to be subset of set ‘B’ if every element of A is an element of set B. If ‘A’ is subset of B, we write A⊆B.

 So, A⊆B if x∈A and x∈B

Eg: If A={1,2} then the subsets of A are ϕ, {1},{2},{1,2}

■Power set of a set:  Let A be a non empty set, then the power set of A is denoted by P(A) and defined by P(A)=2^n, where n=       number of elements in setA.

Eg: If A={1,2,3} then, P(A)=2^n=2^3=8.

OPERATION OF SETS

 Universal Set:

The set X is called Universal set if every set under consideration is a subset of X. Universal set is not a fixed set i.e it varies from situation to situation.

If A={1,2} and B={2,3,4} then   ℕ={set of natural numbers } is called the universal set.

Union of sets.

The union of two sets A and B is defined as the set of all those elements which are in A or B or both. It is denoted by A⋃B.

A⋃B={x:x∈A or x ∈B}

But if  x∉A⋃B then , ⇒x∉A and x ∉B.

Intersection of sets

The intersection of two sets A and B is defined as the set of all those elements which are in both A and B. The intersection of A and B is denoted by A⋂B.

Symbolically, A⋂B={x:x∈A and x ∈B}

But if x∉ A⋂B then ,⇒x∉A or x∉B

 Disjoint Sets

Two sets are said to be disjoint if A⋂B=ϕ.Eg:If A={1,2,3} and B={a,b,c} then we can say that A and B are disjoint sets because A⋂B=ϕ.

 Difference of sets

Let A and B are two sets then the difference of sets A and B is denoted by A-B and defined by :A-B={x:x∈A and x∉B}

Similarly, B-A={x:x∈B and x∉A}

Complement of a set

Let A be the subset of universal set U then, complement of the set A is denoted as 

A^c or A’ or A ̅ = U-A.

i.e A’ ={x:x∈U and x∉A}

Symmetric Difference :

Let A and B be the two sets then symmetric difference between two sets is denoted by  A△B  and defined by

A△B=(A-B)⋃(B-A)