DISCRETE MATH: Determine if two statements are logically equivalent

[VinnyCee]

Homework Statement

Determine whether $$\forall\,x\,(P(x)\,\longleftrightarrow\,Q(x))$$ and $$\forall\,x\,P(x)\,\longleftrightarrow\,\forall\,x\,Q(x)$$ are logically equivalent. Justify your answer.

Homework Equations

$$P\,\longleftrightarrow\,Q$$ is only TRUE when both P and Q are TRUE or FALSE.

The Attempt at a Solution

No, I don't think the two statements are logically equivalent, but I have trouble trying to "justify" my answer.

Set P and Q as always TRUE.

Both statements are equivalent, but if you set P and Q to always FALSE, then the statements are no longer equivalent.

Does this seem logical

 

[matt grime]

Not very. Why not just write things out properly? As in for all z in S then P(z)
is just the same as z in S implies P(z).