Question

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

(B − A) ∪ C = (B ∪ C) − (A − C)

Answer 1

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

A = {1, 2, 5, 7}

B = {2, 7, 8, 9}

C = {1, 5, 8, 7}

Let A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, C = {5, 6, 7, 8}

B – A = {3, 4, 5, 6} – {1, 2, 3, 4}

B – A = {5, 6}

(B – A) ∪ C = {5, 6} ∪ {5, 6, 7, 8}

(B – A) ∪ C = {5, 6, 7, 8}   ......(1)

B ∪ C = {3, 4, 5, 6} ∪ {5, 6, 7,8}

B ∪ C = {3, 4, 5, 6, 7, 8}

A – C = {1, 2, 3, 4} – {5, 6, 7, 8}

A – C = {1, 2, 3 , 4}

(B ∪ C) – (A – C) = {3, 4, 5, 6, 7, 8} – {1, 2, 3, 4}

(B ∪ C) – (A – C) = {5, 6, 7, 8}  ......(2)

From equations (1) and (2)

(B – A) ∪ C = (B ∪ C) – (A – C)