Set Operations, Union-Intersection-Difference-Complement And Properties Of Set Operations And Formulae Of Set Operations

-

The operations which we can perform on the set are set intersection, set union, a difference of two sets and complement, and the properties related to the operation on the set. The operations on set are explained as below:

Union

The union of two sets A and B is a set that consist of all the elements that either belongs to set A or set B or to both. The union is denoted by '⋃' this symbol.  

For example :

suppose that, group A consists of five group members named, Smita, Sakshi, Monali, Akash, and Kiran, and group B consist of four members, named as  Akash, Kiran, Sanika, pakhi. The union of these two groups consists of seven members rather than 9, namely Smita, Sakshi, Monali, Akash, Kiran, Sanika, pakhi.

A={1,2,3}

B={3,4,5,6}

A⋃B ={1,2,3,4,5,6}

B⋃A ={1,2,3,4,5,6}

some properties of operation of union:

  •  A∪B = B∪A                       (Commutative law)
  • A∪(B∪C) = (A∪B)∪C        (Associative law)
  • A ∪ ϕ = A                            (Law of identity element, is the identity of ∪)
  •  A∪A = A                            (Idempotent law)
  • U∪A = U (Law of ∪) ∪ is the universal set.

Notes:

A ∪ ϕ = ϕ ∪ A = A i.e. union of any set with the empty set is always the set itself.

Intersection 

The intersection of two sets A and B  is a set which includes the elements which belongs to both the sets means the set which includes the common element from both the sets.
For example :
By considering the example in union the intersection of the two groups the Intersection of the two groups consists of the group members who are present in both the group.namely, Akash and Kiran.
A={1,2,3,4,5}
B={4,5,6,7,8}
A⋂B={4,5}
The sets whose intersection is an empty set those sets are called as disjoint sets.
If A and B are disjoint sets, then A – B = A and B – A = B.
For example:
A={1,2,3,4}
B={5,6,7,8}
A⋂B={}
Some properties of the operation of intersection

  •  A∩B = B∩A                           (Commutative law)
  •  (A∩B)∩C = A∩ (B∩C)          (Associative law)
  •  ϕ ∩ A = ϕ                                (Law of ϕ)
  •  U∩A = A                                 (Law of ∪)
  •  A∩A = A                                 (Idempotent law)
  •  A∩(B∪C) = (A∩B) ∪ (A∩C) (Distributive law) Here ∩ distributes over ∪
         Also, 
         A∪(B∩C) = (AUB) ∩ (AUC) (Distributive law) Here ∪ distributes over ∩

Notes:

A ∩ ϕ = ϕ ∩ A = ϕ i.e. intersection of any set with the empty set is always the empty
set.

Difference 

the difference of two sets A and B consist of the elements which are present in set A but not in set B
the difference of two sets A and B is denoted as (A-B) or (B-A).Here both the terms A-B and B-A are different. 
For example:
A={1,2,3,4,5,6}
B={4,5,6,7,8,9}
A-B={1,2,3}
B-A={7,8,9}
A-B = A-A⋂B
B-A = B-A⋂B

Complement :

The complement of a set A is a set of elements which are present in universal set but not in set A . The complement of set A is denoted by A' 
For example:
U={1,2,3,4,,5,6,7,8,9,10}
A={1,2,3}
The complement of set A is A' ={4,,5,6,7,8,9,10}
Some properties of complement of set:
  • A ∪ A' = A' ∪ A = ∪         (Complement law)
  • (A ∩ B') = ϕ                      (Complement law)
  •  (A ∪ B) = A' ∩ B'            (De Morgan’s law)
  • (A ∩ B)' = A' ∪ B'            (De Morgan’s law)
  • (A')' = A                            (Law of complementation)
  •  ϕ' = ∪                               (Law of empty set)
  • ∪' = ϕ                                (universal set)

Some important formulae of set operations

If we have two sets:

n(AᴜB) = n(A) + n(B) – n(A∩B)

If we have three sets:

n(A∪B∪C) = n(A) + n(B) + n(C) – n(A∩B) – n(A∩C) – n(B∩C) + n(A∩B∩C)

If we have four sets:

n(A∪B∪C∪D) = n(A) + n(B) + n(C)+ n(D) – n(A∩B) – n(A∩C) – n(B∩C) + n(A∩D) +n(B∩D) +n(C∩D) +n(A∩B∩C) +n(A∩B∩D)+ n(B∩C∩D)+ n(A∩C∩D)- n(A∩B∩C ∩D)

examples on set operations

If set P = {2, 3, 4, 5, 6, 7}, set Q = {0, 3, 6, 9, 12} and set R = {2, 4, 6, 8}.

(i) Find the union of sets P and Q

(ii) Find the union of two set P and R

(iii) Find the union of the given sets Q and R
solution:
given data:
P={2, 3, 4, 5, 6, 7}, 
Q = {0, 3, 6, 9, 12} and
R = {2, 4, 6, 8}.
(I) to find the union of sets P and Q:
P∪Q={0,2,3,4,5,6,7,9,12}
(ii)union of sets P and set R:
P∪R={2, 3, 4, 5, 6, 7,8}
(iii) the union of set Q and R is:
Q∪R={0,2, 3, 4, 6, 8, 9, 12}
 

 What is Set , Definition of Set , Elements of Set , Notation, Representation , Properties  of set and Examples on Sets


Comments

Post a Comment

Popular posts from this blog

Inclusion and Exclusion principle, Venn diagram, examples with solutions, set theory , Mathematics

-
 In this article, you will learn all about the Principle of inclusion and exclusion. the principle of inclusion and exclusion explains how to find the total number of elements that belong to the particular set. principle: for two arbitrary set A and B the principle states as : |A ᴜ B| = |A| + |B| – |A ∩ B| where |A| is the cardinality of the set A. cardinality of the set is the number of elements present in the set. for three sets the principle of inclusion and exclusion is symbolically represented as: |A ∪ B ∪ C| = |A| + |B| + |C| – | A ∩ B|  –  |A ∩ C|  –  |B ∩ C| + |A ∩ B ∩ C| Examples: example 1: Suppose a survey of 100 people asks if they have a cat or dog as a pet.  The results are as follows:  55 answered yes for the cat, 58 answered yes for dog and 20 people checked yes for  both cat and dog. How many people have a cat or a dog? solution: the people who have a cat is denoted by set C the people who have a dog is denoted by  set D |C|...
Read more

Multiset and operations on multisets(union, intersection, addition, difference), solved problems

-
A multiset is a set of elements in which every element can occur more than once. This is the different property of multiset from the sets. a set in which the occurrence of an element can be more than one or can be 0. The multiplicity of the element is the occurrence of the element in the multiset. the  multiplicity of an element can be 1 or more than one or can be 0. the size of the multiset is the sum of all the multiplicities of all the elements.  for example: A= {1,1,1,1,2,2,3,4,4,5} The multiplicity of element '1' in multiset A is 4. The multiplicity of element '2' in multiset A is 2. The multiplicity of element '3' in multiset A is 1. The multiplicity of element '4' in multiset A is 2. The multiplicity of element '5' in multiset A is 1. the size of the multiset A is 10 In sets, the cardinality of the set is the multiplicity of the set. and rest all things the representation of a set and multiset is similar. in multisets A={1,1,1,1,2,2,3,4,4...
Read more

Preposition and logical connectivities in Discrete Mathematics

-
 Preposition A preposition is a declarative statement that is either true or false, but not both. Prepositions contain connectives and variables. The variables in the preposition are denoted by capital letters(A, B,....Z). for example: India is a state                                returns 'false' Maharashtra is a country.               returns 'false' 12-3=9                                            returns 'true' 10-4=7                                          returns  'false' not prepositions    A is greater than 12       x+3=6 go and playout these statements are not prepositions because the statements do not have...
Read more