Express statement in using quantifiers

Question

I can see that it is straight forward when we have an statement like: "For all x, there exists a y, such that f(x,y) = 0", which I can re-write as: "∀ x ∃ y f(x,y) = 0". But, for example, when I have a statement like:

(1) "If x, y ∈ A, then x > y implies f(x) > f(y)"

(2) "Given a > 0, there exists N ∈ N such that for all x ∈ A, n ∈ N implies | fn(x) - f0(x)| < a."

How do I handle these type of cases?

Answer 1

For the first, I would treat the "If x,y ∈ A" part of the statement as "∀ x,y ∈ A" because we want the implication to be true for any x and y we choose from the set A.

For the second, we can interpret that first part of the statement "Given a > 0" as "∀ a > 0", or a bit more formally as "∀ a ∈ R, such that a > 0"