I'm having some issues understanding predicate logic. For example, given the proposition:
These propositions are based on a set of numbers.
$$\forall x \exists y (x \cdot y = x)$$
This means for every $x$ there exists a $y$ that when you multiply you get the same $x$ as an answer. This is true.
However the next statement is
$$\exists y \forall x (x \cdot y = x)$$
This means there exists a $y$ for all $x$'s where when you multiply that $x$ by $y$ you get $x$. I think this is also true (the number 1)
However, I am under the impression that this is wrong and I'm missing something. Could anyone help me understand better?
The first statement says that, for every number $x$, you can find a $y$ that will act as multiplicative identity for it. It could be a different identity for every $x$.
The second statement says that there is a number that acts as multiplicative identity for every other real number. There is indeed such a number, and we call it $1$.
Does that help?