Set Operations

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What are Set Operations?

Set operations can be defined as the operations that are performed on two or more sets to obtain a single set containing a combination of elements from both all the sets being operated upon. There are basically three types of operation on sets in Mathematics; they are:

  • The Union of Sets (∪)

  • The Intersection of Sets (∩)

  • The Difference between Sets (ー)

Let us discuss the operation of set in Math in details:


The Union of Sets (∪)

In Operation set in math, the union of sets can be described as the set that contains all the elements of all the sets on which the union operation is applied. The union of sets can be denoted by the symbol ∪. The set formed by the union of P and Q will contain all the elements of set P and set Q together. The union of sets can be interpreted as:

P ∪ Q = n(P) + n(Q)

Where n(P) represents, the cardinal number of set P and

n(Q) represents the cardinal number of set Q

Let’s take an example:

Set P- {1, 2, 3, 4, 5} and Set Q- {7, 8, 9, 10}

Therefore, P ∪ Q = {1, 2, 3, 4, 5, 7, 8, 9, 10}


The Intersection of Sets (∩)

The intersection of sets is referred to as a set containing the elements that are common to all the sets being operated upon. It is denoted by the symbol ∩. The set that is formed by the intersection of both the sets will contain all the elements that are common in set P as well as Set Q.

Let’s take an example:

Set P= {1, 2, 3, 4} and Set Q= {3, 4, 5, 6}

Then, P ∩ Q = {3, 4}


The Difference Between Sets (-)

The difference of two sets refers to a set that contains the elements of one set that are not present in the other set. It is denoted by the symbol -. Let us say there are two sets P and Q, then the difference between P and Q can be represented as P - Q.

Let’s take an example:

Set P= {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and Set Q= {8, 9, 10}

Therefore, P - Q= {1, 2, 3,4, 5, 6, 7}, here we can see that P - Q contains all those elements that are present in P but not in Q.


Set Operations Venn Diagrams

Venn diagrams are the pictorial representation of sets and set operations. Given below are some set operations Venn diagrams with the explanation.


Venn Diagram for Union of Sets

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The above diagram represents the union of two sets A and B.


Venn Diagram for Intersection of Sets

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The above diagram represents the Intersection of two sets A and B.


Venn Diagram for Difference Between Sets

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The above diagram represents the difference between the set A and B or A - B.


Properties of Set Operations

There are certain properties of set operations; these properties are used for set operations proofs. The properties are as follows:

  • Distributive Property 

  • Commutative Property

  • Distributive Property states that:

If there are three sets P, Q and R then,

P ∩ (Q ∪ R) = (P ⋂ Q) ∪ (P ∩ R)

P ∪ (Q ∩ R) = (P ∪ Q) ∩ (P ∪ R)

  • The Commutative Property states that:

If there are two sets P and Q then,

P ∪ Q = Q ∪ P

P ∩ Q = Q ∩ P


Subset and Powerset


What is a Subset?

A subset can be defined as a set whose elements are the members of another set. The subset is represented by the symbol ⊆. Let us say C is a subset of D then is represented as C ⊆ D. Let us take an example where set D ={1, 2, 3, 4, 5, 6, 7, 8} and set C= { 3, 4, 5, 6, 7}, as we can see here all the elements of set D are present in set C. Therefore, Set C is a subset of Set D.


What is Powerset?

A Powerset is a set of all the subsets. If a set A= {1, 2, 3}, so the subsets of A are 

{}

{1}

{2}

{3}

{1, 2}

{2, 3}

{1, 3} 

{1, 2, 3}

And powerset of A which is denoted by P(A) is { {}, {1}, {2}, {3}, {1, 2}, {2, 3}, {3, 4}, {1, 2, 3}}

The number of element in a power set can be calculated by the formula 2\[^{n}\]. 

FAQs (Frequently Asked Questions)

1. What is the Difference Between Powerset and Subset?

Ans: The difference between subset and powerset is:


A Subset is a set that contains all the elements that are a member of another set. Whereas a powerset is set of all the subsets.  A Subset is denoted by the symbol ⊆, and powerset is denoted by P(). Let us take an example where set A is {1, 2} and set D is {2}. Therefore it can be said that set D is a subset of set A. Now, suppose we have a set A ={1, 2} then subsets of A can be {}, {1}, {2}, {1, 2} and so the powerset of A, i.e. P(A) = {{}, {1}, {2}, {1, 2}}.

2. What is the Complement of a Set?

Ans: In set operations math, consider a universal set U, and A is a subset of U, then the complement of a set A will contain all the elements of set U leaving the elements of set A. The complement of set A is denoted as A’. Let us take an example if Set U, i.e. an universal set which has the following elements U= {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and A is a subset of U, which has the following elements A= {2, 3, 4, 5, 6}, then complement of set A, i.e. A’ ={1, 7, 8, 9, 10}.