# $\forall x\in A(\in\mathcal P(E)),\,P(x)\Leftrightarrow\forall x\in E,\,(x\in A)\wedge P(x)$?

The question is in the title.

When I am working in $$\mathcal P(E)$$, do statements like $$\forall x\in A,\,P(x)$$ translate to $$\forall x\in E,\,(x\in A)\wedge P(x)$$ or to $$\forall x\in E,\,(x\in A)\implies P(x)$$?

• At the title's beginining, did you mean this instead: $\forall x\in A(\subset E),\,P(x)?$ Dec 4 '21 at 15:23
• @ryang thank you! it should have been $\in$. Dec 4 '21 at 15:33
• @ryang $\mathcal P(E)$ is the set of all subsets of $E$. Dec 4 '21 at 16:35
• Oh, power set. Why not just say subset of E instead of element of E's power set. Dec 4 '21 at 16:44
• @ryang the notes that i am using usually employ $\mathcal P(E),\,(\mathcal P(E))^2$ etc ... Dec 4 '21 at 16:50

$$\forall x[x\in A\Rightarrow P(x)]$$
$$\exists x[x\in A \wedge P(x)]$$.