For each of 5-21 prove each statement that I true and find a counterexample for each statement that is false. Assume all sets are subsets of a universal set U.
For all sets A and B, P ( A ) ∪ P ( B ) ⊆ P ( A ∪ B ) .

To Prove:

For each prove each statement that is true and find a counterexample for each statement that is false. Assume all sets are subsets of a universal set U.

For all sets A and B, P(A)∪P(B)⊆P(A∪B).

Given information:

Let  A and B  be any set

Concept used:

∪:Union of sets∩:Intersection of sets

⊆: subset of set

Calculation:

Consider the following statement,

P(A)∪P(B)⊆P(A∪B)

The objective is to verify whether the statement is true or not.

Recall, the definition for a power set.

A Power Set is a set of all the subsets of a set.

Let A and B are two sets, then the Power Sets of A and B are defined by,

P(A)={X|X is a subset of A} and P(B)={X|X is a subset of B}

The sets in P(A)∪P(B) contains only sets that are subsets of either A and B, and P(A∪B) contains the elements of both the sets A and B.

Let A and B are two subsets of a universal set U then