Hello I have a question about mathematics: when I have a variable x, if I have a statement: $\forall x P(x) \land \forall x Q(x)$, is it implied that I am testing the same x in both P(x) and Q(x)?
in other words: is the following true: $\forall x P(x) \land \forall Q(x) \leftrightarrow \forall x [ P(x) \land Q(x)$
I suppose the best way to put my question is: When I have two quantifiers referring to the same variable, can I assume they will always be referring to exactly the same variable, or do I treat them as two different variables which happen to have the same symbol of reference?
Well, I ended up finding an explanation to my problem. here is the relevant question if anyone happens to be interested later on:
Why is: $$\forall x(p(x)\vee q(x))\not\equiv\forall x(p(x))\vee \forall x(q(x))$$
Where as: $$\forall x(p(x)\wedge q(x))\equiv\forall x(p(x))\wedge \forall x(q(x))$$
And in the same manner, why is: $$\exists x(p(x)\vee q(x))\equiv\exists x(p(x))\vee \exists x(q(x))$$
But: $$\exists x(p(x)\wedge...
Why is: $$\forall x(p(x)\vee q(x))\not\equiv\forall x(p(x))\vee \forall x(q(x))$$
Where as: $$\forall x(p(x)\wedge q(x))\equiv\forall x(p(x))\wedge \forall x(q(x))$$
And in the same manner, why is: $$\exists x(p(x)\vee q(x))\equiv\exists x(p(x))\vee \exists x(q(x))$$
But: $$\exists x(p(x)\wedge...
