What are Set Operations?

Till now the students have dealt with mainly four basic operations of mathematics i.e. addition, subtraction, multiplication and division. These operations are mainly applied to two or more numbers to obtain a result that is a combination of these numbers. For example, when we apply the operation of addition on two numbers suppose 7 and 2, we get the number 9. Likewise, set operations are a group of operations that are applied on two or more sets that combines them and results in a single set. The set operations consist of three types of operations namely the union of sets (U), the intersection of sets (⋂) and the difference between sets (-). Let us understand all the set operations with suitable examples:

Union of Sets

Suppose A and B are sets consisting of elements. Let's say set A = {4, 5, 6, 2, 1} and set B = {7, 8, 9, 0}.

A⋃B is read as ‘A union B’, that means ‘union’ is denoted as ‘⋃’.

Therefore, A ⋃ B =  {4, 5, 6, 2, 1} ⋃  {7, 8, 9, 0}.

       = {4, 5, 6, 2, 1, 7, 8, 9, 0}

Hence, we see that the union of sets A and B consists of all the elements that were in set A and set B respectively.

Intersections of Sets

Suppose A and B are sets consisting of elements. Let's say set A = {8, 9, 5, 4, 6, 2} and set B = {5, 2, 3, 1, 9}.

A∩B is read as ‘A intersection B’, that means ‘intersection’ is denoted as ‘∩’.

Therefore, A⋂B = {8, 9, 5, 4, 6, 2}⋂{5, 2, 3, 1, 9}.

    = {5, 2, 9}

Hence, when we apply intersection on two sets it gives us the elements that are common or are a part of both the sets.

Difference of Sets

Suppose A and B are two sets consisting of elements. Let’s say set A = {5, 6, 8, 9, 0} and B = { 9, 6, 0, 7, 3}

A - B is read as ‘A minus B’, that means the difference or minus is denoted as ‘-’.

Therefore, A - B =  {5, 6, 8, 9, 0} - { 9, 6, 0, 7, 3}

    = {5,8}

We can see that the difference of two sets A and B results in the set of elements that are a part of set A but are not in set B.

Similarly, B - A = { 9, 6, 0, 7, 3} - {5, 6, 8, 9, 0} 

  = {7, 3}

Here also, B - A results in the set of elements that are a part of set B but are not in set A.

Solved Examples

1. If A = {6, 9, 8, 1}, B = {1, 7, 5, 2, 9}, C = {7, 6, 0, 1} and D = {0, 1}. Find:-

1. A ∩ B

2. B ∩ C

3. A ∩ C

4. B ∩ D

5. A ∩ D

6. A ∩ (B U C)

7. A ∩ (B U D)

Answer - (a) A ∩ B  = {6, 9, 8, 1} ∩  {1, 7, 5, 2, 9}

                            = {1, 9}

(b) B ∩ C =  {1, 7, 5, 2, 9} ∩ {7, 6, 0, 1}

               = {1, 7}

(c) A ∩ C = {6, 9, 8, 1}⋂ {7, 6, 0, 1} 

                = {1, 6}

(d) B ∩ D = {1, 7, 5, 2, 9} ⋂  {0, 1}

                = {1}

(e) A ∩ D = {6, 9, 8, 1} ⋂ {0, 1}

                = {1}

(f) A ∩ (B U C) = {6, 9, 8, 1} ⋂ ({1, 7, 5, 2, 9} U {7, 6, 0, 1})

                        = {6, 9, 8, 1} ∩ {1, 7, 5, 2, 9} U  {6, 9, 8, 1} ⋂ {7, 6, 0, 1}

                        = {1, 9} U {1, 6}

                        = {1, 9, 6}

(g) A ∩ (B U D) = {6, 9, 8, 1} ∩ ({1, 7, 5, 2, 9} U {0, 1})

                         = {6, 9, 8, 1} ∩ {1, 7, 5, 2, 9} U {6, 9, 8, 1} ∩ {0, 1}

                         = {1,9} U {1}

                         = {1, 9}