Question

By taking suitable sets A, B, C, verify the following results:

C − (B − A) = (C ∩ A) ∪ (C ∩ B')

Answer 1

To prove the following results let us take U = {1, 2, 5, 7, 8, 9, 10}

Let A = {1, 2, 3) , B = {2, 3, 4) , C = {3, 4, 5}

B – A = {2, 3, 4) – {1, 2, 3}

B – A = {4}

C – (B – A) = {3, 4, 5} – {4}

C – (B – A) = {3, 5}   .....(1)

C ∩ A = {3, 4, 5} ∩ { 1, 2, 3)

C ∩ A = {3}

B’ = {1, 5}

C ∩ B’ = {3, 4, 5} ∩ {1, 5}

C ∩ B’ = {5}

(C ∩ A) ∪ (C∩B’) = {3} ∪ {5}

(C∩A) ∪ (C ∩ B’) = {3, 5}   ......(2)

From equations (1) and (2)

C – (B – A) = (C ∩ A) u (C ∩ B’)