Proving P(A) Union P(B) is a Subset of P(A Union B)

[DPMachine]

Homework Statement

Prove $$P(A) \cup P(B) \subseteq P(A \cup B)$$

Homework Equations

The Attempt at a Solution

I started out by assuming that $$A = \left\{a\right\}$$ and $$B=\left\{b\right\}$$.

So then $$P(A) \cup P(B) = \left\{\left\{a\right\},\left\{b\right\},null\right\}$$ and $$P(A \cup B) = \left\{\left\{a\right\},\left\{b\right\},\left\{a,b\right\},null\right\}$$

So I can conclude that $$P(A) \cup P(B) \subseteq P(A \cup B)$$

How does that sound?

 

[n!kofeyn]

When you assume that A={a} and B={b}, you are assuming that A and B are both singleton sets. You want to prove the relation for any sets A and B.

When proving one set is a subset of another, say X is a subset of Y, then you let x be in X and show x is in Y. So let $x\in P(A)\cup P(B)$, and then show $x\in P(A\cup B)$.

 

[DPMachine]

When you assume that A={a} and B={b}, you are assuming that A and B are both singleton sets. You want to prove the relation for any sets A and B.

When proving one set is a subset of another, say X is a subset of Y, then you let x be in X and show x is in Y. So let $x\in P(A)\cup P(B)$, and then show $x\in P(A\cup B)$.

Okay, so let $$x\in P(A)\cup P(B)$$.
Then $$x\in P(A)$$ or $$x\in P(B)$$... which means $$x\subseteq A$$ or $$x\subseteq B$$?

So then $$x\subseteq (A\cup B)$$. Am I going in the right direction?

 

[Landau]

Very good, the conclusion now follows.