Question

Prove the following set identity using logical equivalences: A ∪ (B - A) = A ∪ B.\

(Hint: Insert a table with 2 columns and 8 rows.)

Answer 1

A (B - A) = A B

L.H.S. = A (B - A)

= A (B ∩ A^C) {Set Difference Law, A - B = A ∩ B^C }

= (A B) ∩ (A A^C) {Distributive Law, A (B ∩ C) = (A B) ∩ (A C)}

= (A B) ∩ T {Complement Law, A A^C = T }

= (A B) {Identity Law, A ∩ T = A }

= R.H.S.

As, L.H.S. = R.H.S.. It is clear that A (B - A) = A B is logically equivalent.

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