Solutions – Union and Intersection of two Sets

1.If X and Y are two sets such that X∪Y has 40 elements, X has 22 elements and Y has 30 elements, how many elements does X∩Y have?

Solution: Given that n(X∪Y) = 40,  n(X) = 22, and   n(Y) = 30, n(X∩Y)= ?

By using the formula

 n(X∪Y) = n(X) + n(Y) – n(X∩Y)

              put the values in formula 

        n(X∪Y) = 40,  n(X) = 22, and   n(Y) = 30

                             n(X∩Y)= ?

                        40 = 22 + 30 – n(X∩Y)

                        40 = 52 – n(X∩Y)

                        40 – 52 = – n(X∩Y)

                      – 12 = – n(X∩Y)

                  n(X∩Y) = 12      Solution.

2. If X and Y are two sets such that X∪Y has 38 elements, X has 17 elements and Y has 23 elements, how many elements does X∩Y have?

Solution: Given that n(X∪Y) = 38,  n(X) = 17, and   n(Y) = 23, n(X∩Y)= ?

By using the formula

 n(X∪Y) = n(X) + n(Y) – n(X∩Y)

              put the values in formula 

        n(X∪Y) = 38,  n(X) = 17, and   n(Y) = 23

                             n(X∩Y)= ?

                        38 = 17 + 23 – n(X∩Y)

                        38 = 40 – n(X∩Y)

                        38 – 40 = – n(X∩Y)

                      – 2 = – n(X∩Y)

                         n(X∩Y) = 2      Solution. 

3. If P and Q are two sets such that P∪Q has 18 elements, P has 8 elements and Q has 15 elements, how many elements does P∩Q have?

Solution: Given that n(P∪Q) = 18,  n(P) = 8, and   n(Q) = 15, n(P∩Q)= ?

By using the formula

 n(X∪Y) = n(X) + n(Y) – n(X∩Y)

              n(P∪Q) = n(P) + n(Q) – n(P∩Q)

              put the values in formula 

        n(P∪Q) = 18,  n(P) = 8, and   n(Q) = 15,

                             n(P∩Q)= ?

                        18 = 8 + 15 – n(P∩Q)

                        18 = 23 – n(P∩Q)

                        18 – 23 = – n(P∩Q)

                      – 5 = – n(P∩Q)

                         n(P∩Q) = 5      Solution 

4. In a class of 100 students, 35 like english and 45 like hindi. 10 like both. How many like either of them and how many like neither?

Solution: Total number of students U = 100
Number of english students n(E) = 35

Number of hindi students n(H) = 45

Number of students who like both = 10

Number of students who like either of them,

n(E∪H) = n(E) + n(H) – n(E∩H)

               n(E∪H) = 45 + 35 -10

               n(E∪H)   = 70

Number of students who like neither of them,

                            = U – 100

                            = 100 – n(E∪H)

                            = 100 – n(E∪H)

                            = 100 -70

                            = 30 

Number of students who like neither of them = 30
The solution is there in venn diagram.