UNION AND INTERSECTION OF SETS

Example 1 :

List :

(i)  set C

(ii)  set D

(iii)  set U

(iv)  set CnD 

(v)  set CUD

Solution :

(i)  set C

The elements in set C  =  {1, 3, 7, 9}

(ii)  set D  

The elements in set D  =  {1, 2, 5}

(iii)  set U

The universal set with all the elements in set

U  =  {1, 2, 3, 4, 5, 6, 7, 8, 9}

(iv)  set CnD 

The common elements in sets C and D is

C n D  =  {1}

(v)  set C U D

All the elements in sets C and D is

CUD  =  {1, 2, 3, 5, 7, 9}

Example 2 :

Find :

(i)   n(C)

(ii)  n(D)

(iii)  n(U)

(iv)  n(CnD)

(v)  n(CUD)

Solution :

(i)  n(C)

The number of elements in set C is 4.

So, n(C)  =  4

(ii)  n(D)

The number of elements in set D is 3

So, n(D)  =  3

(iii)  n(U)   

The number of elements in universal set U is 9

n(U)  =  9

(iv)  n (CnD)

The number of common elements in sets C and D is 1.

n(CnD)  =  1

(v)  n (CUD)

The number of all elements in sets C and D is 6.

n(CUD)  =  6

Example 3 :

List :

(i)  set A

(ii)  set B

(iii)  set U

(iv)  set AnB

(v)  set AUB

Solution :

(i) A  =  {2, 7}

(ii) B  =  {1, 4, 6, 2, 7}

(iii) U =  {1, 2, 3, 4, 5, 6, 7, 8}

(iv) AnB  =  {2,7}

(v) AUB  =  {1, 2, 4, 6, 7}

Example 4 :

Find :

(i)  n(A)

(ii) n(B)

(iii) n(U)

(iv) n(AnB)

(v)  n(AUB)

Solution :

(i)   n(A)

The number of elements in set A  =  2

(ii)  n(B)

The number of elements in set B  =  5

(iii)  n(U)   

The number of elements in universal set U  =  8

(iv) n(AnB)

The number of common elements in sets A and B is 2

n(AnB)  =  2

(v)  n(AUB)

The number of all elements in sets A and B is 5.

So, n(AUB)  =  5

Example 5 :

Consider  U  =  {x | x ≤ 12, x Є Z⁺}

A  =  {2, 7, 9, 10, 11} and B  =  {1, 2, 9, 11, 12}

Show these sets on a Venn diagram.

Solution :

Example 6 :

List the elements of :

(i)  AnB

(ii) AUB

(iii)  B’

Solution :

(i) AnB  =  {2, 9, 11}

(ii)  AUB  =  {1, 2, 7, 9, 10, 11, 12}

(iii)  B’

The elements which do not belong to set B is

B’  =  {3, 4, 5, 6, 7, 8, 10}

Example 7 :

Find :

(i)  n(A)

(ii)  n(B’)

(iii)  n(AnB)

(iv)   n(AUB)

Solution :

(i)   n(A)  =  5

(ii) The number of elements which do not belong to set B is B’  =  7

(iii)  n(AnB)  =  3

(iv)  n(AUB)  =  7

Example 8 :

If A is the set of all factors of 36 and B is the set of all factors of 63,

Find : (a)   AnB     (b)   AUB

Solution :

Given, A is the set of all factors of 36

So, set A  =  {1, 2, 3, 4, 6, 9, 12, 18, 36}

B is the set of all factors of 63

So, set B  =  {1, 3, 7, 9, 21, 63}

(a)  AnB  =  {1, 3, 9}

(b)  AUB  =  {1, 2, 3, 4, 6, 7, 9, 12, 18, 21, 36, 63}

Example 9 :

If X  =  {A, B, D, M, N, P, R, T, Z} and

Y  =  {B, C, M, T, W, Z}

Find :  (a)   XnY     (b)   XUY

Solution :

Given, X  =  {A, B, D, M, N, P, R, T, Z} and

Y  =  {B, C, M, T, W, Z}

(a)   XnY  =  {B, M, T, Z}

(b)   XUY  =  {A, B, C, D, M, N, P, R, T, W, Z}

Example 10 :

If U  =  {x| x ≤ 30, x Є Z⁺}

A  =  {factors of 30} and B  =  {prime numbers ≤ 30}

Find :

(a)  (i)   n (A)   (ii)  n (B)   (iii)  n (A n B)   (iv)  n (A U B)

(b)  Verify that n(AUB)  =  n(A) + n(B) – n(AnB)

Solution :

Given, A  =  {1, 2, 3, 5, 6, 10, 15, 30}

B  =  {2, 3, 5, 7, 11, 13, 17, 19, 23, 29}

(i)   n(A)  =  8

(ii)  n(B)  =  10

(iii)  n (AnB)  =  3

(iv)  n (AUB)  =  15

(b)  n(AUB)  =  n(A) + n(B) – n(AnB)

15  =  8 + 10 – 3

15  =  18-3

15  =  15

Hence it is verified.

Example 11 :

Of 75 people with cell phones, 42 take pictures with their cell phone and 36 use their cell phone to take pictures and send text messages. How many use their phones to send text messages ?

Solution :

Let A and B be the number of people who take pictures and sending messages.

n(A) = 42

n(A n B) = 36

Total number of people = n(AUB) = 75

n(AUB) = n(A) + n(B) - n(A n B)

75 = 42 + n(B) - 36

75 = 6 + n(B)

n(B) = 75 - 6

n(B) = 69

Number of people who use phones and send text messages is 69.

Example 12 :

Find two sets A and B such that A U B = {1, 2, 3, 4, 5 } and A n B = {2}

Solution :

A U B = {1, 2, 3, 4, 5 }

A n B = {2}

A = {1, 3, 4}

B = {5}

Kindly mail your feedback to v4formath@gmail.com

We always appreciate your feedback.