In 30-40, construct an algebraic proof for the given statement, Cite a property from Theorem 6,2,2 for every step.

For all sets A and B, ( A ∩ B ) c ∩ A = A − B .

To prove:

For all sets A and B, A−(A∩B)=A−B

Given information

(A∪B)−C=(A−C)∪(B−C)

Concept used:

(A∪B)−C=(A−C)∪(B−C)

Cite a property from Theorem 6.2.2 for every step of the proof.

Solution let A,B and C be any sets. Then

(A∪B)−C=(A∪B)∩Cc                  by the set difference law=Cc∩(A∪B)                  by the commutative law for ∩=(Cc∩A)∪(Cc∩B)      by the distributive law=(A∩Cc)∪(B∩Cc)      by the commutative law for ∩=(A−B)∪(B−C)           by the set difference law

Calculation:

Consider that A,B be any two sets