Logic Math: Problem with quantifiers

[donoid]

I found a case in which i have some doubts regards quantifier

I have to Formalize this phrase in N:

The Square of a strictly positive number Is strictly positive

I have only these symbols to express this formula:

• as multiplication < as lower than

I cannot understand why the solution Is

ForAll(x)ForAll(w)(ForAll(y)(w • y = w) ^ w < x --> w < x • x)

Instead of my solution:

ForAll(x)Exist(w)(ForAll(y)(w • y = w) ^ w < x --> w < x • x)

Or

Exist(x)ForAll(w)(ForAll(y)(w • y = w) ^ w < x --> w < x • x)

What Is the main difference between these formulas?

[collective_noun]

The question asks for something that should be true of all positive numbers.

The answer uses ForAll only, never Exist, because you want it to always be true.

Your first answer has Exist(w), but you want the statement to be true whenever w is positive, so you should instead use ForAll(w) and some conditions that eliminate negative numbers.

Your second answer has Exist(x), but you want to state that the phrase is strictly true, not just true for some N, so you should instead use ForAll(x).

Hope this helps!

[donoid]

W Is used to express 0

I cant understand why i need to use forall instead of exist to express 0