Question

For any two sets, Prove that P(A) U P(B) subset P(A U B) and P (A U B) is not necessarily a subset of P(A) U P(B)

Answer 1

Step-by-step explanation:

Suppose X ∈P(A)∪P(B). By definition of union, this means X ∈ P(A) or X ∈ P(B). By definition of power sets X ⊆ A or X ⊆ B Case 1. Suppose X ⊆ A. Then X ⊆ A∪B, and this means X ∈P(A∪B). Case 2. Suppose X ⊆B. Then X ⊆ A∪B, and this means X ∈P(A∪B). By case 1 and 2, X ∈P(A∪B). Thus X ∈P(A)∪P(B) implies X ∈P(A∪B), and therefore P(A)∪P(B)⊆ P(A∪B).