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Mathematics | Set Operations (Set theory)

Union
Union of the sets A and B, denoted by A ∪ B, is the set of distinct element belongs to set A or set B, or both.

AUB

Above is the Venn Diagram of A U B.

Example : Find the union of A = {2, 3, 4} and B = {3, 4, 5};
Solution : A ∪ B = {2, 3, 4, 5}.

 



Intersection
The intersection of the sets A and B, denoted by A ∩ B, is the set of elements belongs to both A and B i.e. set of the common element in A and B.

AinterB

Above is the Venn Diagram of A ∩ B.

Example: Consider the previous sets A and B. Find out A ∩ B.
Solution : A ∩ B = {3, 4}.

 

Disjoint
Two sets are said to be disjoint if their intersection is the empty set .i.e sets have no common elements.

AdijointB

Above is the Venn Diagram of A disjoint B.

For Example
Let A = {1, 3, 5, 7, 9} and B = { 2, 4 ,6 , 8} .
A and B are disjoint set both of them have no common elements.

 



Set Difference
Difference between sets is denoted by ‘A – B’, is the set containing elements of set A but not in B. i.e all elements of A except the element of B.

A-B

Above is the Venn Diagram of A-B.

 

Complement
Complement of a set A, denoted by

A^complement

, is the set of all the elements except A. Complement of the set A is U – A.

Acomplemnt

Above is the Venn Diagram of Ac




    Formula:

  1. Acup B =n(A) + n(B) - n(Acap B)

  2. A-B=Acap ar{B}


    Properties of Union and Intersection of sets:

  1. Associative Properties: A ∪ (B ∪ C) = (A ∪ B) ∪ C and A ∪ (B ∩ C) = (A ∩ B) ∩ C
  2. Commutative Properties: A ∪ B = B ∪ A and A ∩ B = B ∩ A
  3. Identity Property for Union: A ∪ φ = A
  4. Intersection Property of the Empty Set: A ∩ φ = φ
  5. Distributive Properties: A ∪(B ∩ C) = (A ∪ B) ∩ (A ∪ B) similarly for intersection.

  6. Example : Let A = {0, 2, 4, 6, 8} , B = {0, 1, 2, 3, 4} and C = {0, 3, 6, 9}. What are A ∪ B ∪ C and A ∩ B ∩ C ?

    Solution: Set A ∪ B &cup C contains elements which are present in at least one of A, B, and C.

    A ∪ B ∪ C = {0, 1, 2, 3, 4, 6, 8, 9}.

    Set A ∩ B ∩ C contains an element which is present in all the sets A, B and C .i.e { 0 }.

     

    See this for Set Theory Introduction.

     

    Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above



This article is attributed to GeeksforGeeks.org

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