Question

Let A = {1, 3, 5, 6, 7}, B = {4, 6, 7, 9} then verify the following identities.
(i) n (A -B) = n (A) - n (A $\cup$ B)
(ii) n (A$\cup$B) = 9, n (A$\cap$B) = 2, find n (B).

Answer 1

Given: A = {1,3,5,6,7} and B = {4,6,7,9}

(i) (A $-$ B) = {1,3,5}
i.e., n(A $-$ B) = 3 ...(1)

Also, n(A) = 5 and (A$\cap$B) = {6,7}
i.e., n(A$\cap$B) = 2 ...(2)
Now, n(A) $-$ n(A$\cap$B) = 5 $-$ 2 = 3 ...(3)

From (1) and (3), we get:
n(A $-$ B) = n(A) $-$ n(A$\cap$B)

(ii) (A$\cup$B) = {1,3,4,5,6,7,9}
i.e., n(A$\cup$B) = 7 ...(4)
From (2), n(A$\cap$B) = 2

Also, n(B) = 4
Now, n(A) + n(B) $-$ n(A$\cap$B) = 5 + 4 $-$ 2 = 7 ...(5)

From (4) and (5), we have:
n (A$\cup$B) = 7 = n (A)+ n(B) $-$ n(A$\cap$ B)