Consider the following statement. For all sets A and B, (A ∪ Bc) − B = (A − B) ∪ Bc.

Question

An algebraic proof for the statement should cite a property from Theorem 6.2.2 for every step, but some reasons are missing from the proposed proof below. Indicate which reasons are missing. (Select all that apply.)

Let any sets A and B be given. Then

(A ∪ Bc) − B	=	(A ∪ Bc) ∩ Bc		by the set difference law	(1)
=	(Bc ∩ A) ∪ (Bc ∩ Bc)		by the distributive law	(2)
=	(Bc ∩ A) ∪ Bc		by the idempotent law for ∪	(3)
=	(A − B) ∪ Bc		by the set difference law	(4)

A. The absorption law is needed between steps (2) and (3).

B. The commutative law is needed between between steps (1) and (2).

C. The commutative law is needed between between steps (3) and (4).

D. The complement law is needed between steps (2) and (3).

E. The double complement law is needed between steps (3) and (4).

Answer 1

(A U ~B) - B = <---- given

(A U ~B) intersect ~B <---- set difference law

(A intersect ~B) U (~B intersect ~B) <--- distributive

(A intersect ~B) U ~B <--- absorption

(A - B) U ~B <--- set difference law in reverse