# DISCRETE MATH: Determine if two statements are logically equivalent

1. Jan 18, 2007

### VinnyCee

1. The problem statement, all variables and given/known data

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.

2. Relevant equations

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

3. 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

2. Jan 18, 2007

### 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).