SETS & VENN DIAGRAM

SETS

Objects :- Everything in this universe, whether living or non living is called an object.

Well-defined collection of objects :- A collection of objects is said to be well defined if it is possible to tell beyond doubt about every object of the universe, whether it is there in our collection or not.

Set :- A well defined collection of object is called a set.

The objects in a set are called its members or elements. We usually denote set by capital letters A,B etc.

Representation of sets :- 

  1. Roster Form(List Form) :- Here a set is represented by listing all its elements in curly brackets and separating them by commas. Ex: {2,3,4}.
  2. Rule Form(Set builder form) :- A set is represented by stating all the properties which are satisfied by the elements of the set and not by any other elements outside the set. Ex: {x:x= 4n+1,1<n<7,n∈N}

Finite Set :- A set is called a finite set, if it contains only finite number of elements. Ex: {1,2,3,4….10}.A set not containing finite number of elements is called an infinite set. Ex: {3,6,8,12,…….}

Universal Set :- A set X is called a uniersal set if every set under consideration is a subset of X.



Null Set :- A set containing no elements is called a null set. A null set is also called as empty set or void set and denoted by φ. Ex:{x:2<x<3,x∈N}

Singleton Set :- A set containing exactly one element is called a singleton set. Ex:{x:2<x<=3,x∈N}

Equivalent set :- Two sets A and B are said to be equivalent if they have equal number of elements.

Equal Sets :- Two sets are said to be equal sets if each element of one set is in other set and vice versa.

Subset :- A set A is said to be subset of B, if every element of A is an element of B.

NOTE :

Number of proper set of set consisting n elements is 2n-1

Power set :- The set of all subsets of A set is called its power set and is denoted by P(A).

NOTE :

Number of elements of power sets consisting n elements is  2n

Proper subset :- A set A is said to be a proper subset of set B if every element of A is an element of B and there is atleast one element in B which is not an element of A.

If A is a proper subset of B, then we represent it as (A⊂B).

NOTE :

  • Number of proper subsets of set consisting n elements is 2n-1
  • Number of non-empty subsets of set containing n elements is 2n-1
  • Number of non empty proper subsets of set consisting n elements is 2n-2

Union of sets :- The Union of two sets A and B is defined as sets of all those elements which are either in A or B both and denoted by A∪B

A∪B = {x:x∈A or  x∈B}

Intersection of sets :- The intersection of two sets A and B is defined as the set of all those element which are in both A and B and denoted by A∩B

A∩B ={x:x∈A and  x∈B}

Disjoint sets :- Two sets A and B are said to be disjoint if there is no element which is in both A and B. If A and B are disjoint sets then A∩B = φ

Difference of sets :- The difference of two sets A and B is set of all those elements of  A which are not in B. The difference of A and B is denoted by A-B.

A-B = {x:x∈A and x∉B}

Symmetric difference of sets :- The Symmetric Difference of sets A and B is defined as the union of the sets (A-B) and (B-A).

AΔB = {x:x ∈ A-B or x ∈ B-A} = (A-B)∪(B-A) = (A∪B)-(B∪A)

Complement of sets :- If A is a subset of universal set X, then the complement of A w.r.t to X is defined as a set of all those elements of X which are not in A and is denoted by Ac.

A= {x : x∈A and  x∉A}

Important Results: If A,B and C are finite sets and U be the finite universal set then

Sets



VENN DIAGRAM

In order to understand and illustrate the relationship among sets, we represent them pictorially by means of diagrams known as venn diagrams.

In venn diagrams the universal set U is represented by a rectangular region and each of its subsets is represented by a closed bounded figure.

Sets