Determine whether All X ( P(X) If only if Q(X)) and All X P(X) If only if All X Q(X) are logically equivalent.
Thank you very much
Seems to be you want to check whether $\displaystyle \forall x\left(P(x)\Longleftrightarrow Q(x)\right)\equiv \left(\forall x\,P(x)\Longleftrightarrow \forall x\,Q(x)\right)$ .
Well, let $\displaystyle \mathbb{Z}$ be the universe from where we get our objects ,and let P(x) = the absolute value of x is x itself (i.e., P(x) iff $\displaystyle |x|=x$ ,and let Q(x)= x is a non-negative integer (i.e., Q(x) iff $\displaystyle x\geq 0$) , then:
$\displaystyle \forall x\left(P(x)\Longleftrightarrow Q(x)\right)$ means: for any integer x, $\displaystyle |x|=x\,\,\,iff\,\,\,x\leq 0$ , which is true, whereas
$\displaystyle \left(\forall x\,P(x)\Longleftrightarrow \forall x\,Q(x)\right)$ means: for any integer x we have $\displaystyle |x|=x\,\,\,iff$ for any integer x, $\displaystyle x\geq 0$ , which is false.
If I didn't commit some logical mistake above () then the answer is: no, they aren't logical equivalent.
Tonio
No. Consider P(x) = "x=0", and Q(x)="x>0"
Clearly For all x, P(x) iff Q(x) is false. But, "for all x, P(x)" is false, and "for all x, Q(x)" is false. Then "for all x, P(x) iff for all x, Q(x)" is true.
The converse is true though, because assume "for all x, P(x) iff Q(x)". Then if there there is a x where P(x) is false then Q(x) is also false, so "for all x, P(x) iff for all x, Q(x)" is true as each universal is false. Otherwise, P(x) is always true, as is Q(x); result follows.