Hello SkinnerCorrect. For example, if x is a real number, then for all x, there exists a y such that y > x. In other words, there is no largest real number.

Again, I think you are interpreting this correctly; the value of y is chosen before the values of x. For example, if x and y are integers, there exists a y such that for all x, xy = x. (The integer is, of course, y = 1.)

This is rather harder to come up with a simple example. But you just combine the ideas in the first two examples; it means, of course, what it says, reading from left to right: For all values of x, there exists a value of y such that, for all values of z, such-and-such a statement is true. The following example works, but it may seem a little trivial.

Suppose we are considering elements of the set of non-negative integers: . Then

Can you see what this value of is?

Does that help?

Grandad