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. A\cup B =n(A) + n(B) - n(A\cap B)

  2. A-B=A\cap \bar{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.

     

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