What do we mean when we say "for all n where n is an element of..."

Question

When we write "for all n where n is an element of A, P(n) is true." do we mean for all n, if n is in A then Pn or for all n, n is in A and Pn?

Answer 1

Bounded quantifiers are $\exists a\in A.P(a)$ and $\forall a\in A.P(a)$ are abbreviations which tell us that we are interested in the elements of $A$ rather than those of the rest of the universe. The expanded versions are: $$\exists a\in A.P(a)\equiv\exists a(a\in A\land P(a))\\ \forall a\in A.P(a)\equiv\forall a(a\in A\rightarrow P(a))$$

Note that the second interpretation that you gave (for all $n$, $n\in A$ and $P(n)$) is not a very good one, because it means that $A$ is equal to the universe of discourse, and that all elements satisfy $P$.

Answer 2

It means the first...symbolically,

$$\forall n\Big(n \in A \rightarrow P(n)\Big)$$ In words, for all $n$, if $n$ is an element of $A$, then $P(n)$ is true.