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.