Question

g Consider the following statement. For all sets A and B, (A − B) ∪ (A ∩ B) = A. Construct an algebraic proof for the statement. Cite a property from Theorem 6.2.2 for every step.

Answer 1

Answer:

(A − B) ∪ (A ∩ B) = A

Step-by-step explanation:

we will use below conditions

Let A and B be the two sets

Distributive property:-

i) A U (B ∩ C) =( AUB)∩(AUC)

ii) A ∩ (BUC) = (A ∩ B) U (A ∩ C)

we will use another conditions also iii)  A-B =A ∩ B¹

now Given (A − B) ∪ (A ∩ B) = (A ∩ B¹) ∪ (A ∩ B) ( from (iii)

                                             = A ∩ (BUB¹)   ( from (ii)

                                             = A ∩ U ( see the diagram)

                                             = A

∴ (A − B) ∪ (A ∩ B) = A