# Thread: Is this set theory statement true?

1. ## Is this set theory statement true?

I think it is, but if someone could explain why or why not, it would be greatly appreciated.

{4} ∈ P{6,4,3,2,5}

Many thanks! I'm just getting my head around set theory.
(The P is a power set symbol by the way!)

2. ## Re: Is this set theory statement true?

$A\in P(B)$ means the same thing as $A\subseteq B$. Here, it is clearly true that $\{4\}\subseteq\{6,4,3,2,5\}$.

3. ## Re: Is this set theory statement true?

Your help is much appreciated, thank you. I never realised that it meant the same thing.

4. ## Re: Is this set theory statement true?

Originally Posted by emakarov
$A\in P(B)$ means the same thing as $A\subseteq B$.
This is almost always true, but not quite. You can apply $A\subseteq B$ in some cases when B is a proper class (and even if A is a proper class), but you can't take the power set of a proper class so $A\in P(B)$ is not a valid statement. For example, it makes sense to say that the class of all ordinals form a subset of the class of all sets, and I can describe this in first-order logic without getting into paradoxes. However, it doesn't make sense to refer to the power set of the class of all sets (such a set does not exist).

The important idea here is that the power set of a set X, is the collection of all X's subsets. So the members of P(X), when P(X) is a set, are precisely those things that are subsets of X.