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 − A ) = A ∪ B .

To prove:

For all sets A and B, 

A∪(B−A)=A∪B.

Given information

(A∪B)−C=(A−C)∪(B−C).

Concept used:

(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−C)∪(B−C)   by the set difference law.

Calculation:

The objective is to construct an algebraic proof for the following statement

A∪(B−A)=A∪B

Consider

A∪(B−A)={x/x∈A o