Stating whether the stament that uses predicate logic is true in certain domains?

Question

Let Z = {...,−2,−1,0,1,2,...} be the domain of integers and N the naturals (i.e. non-negative integers). The predicate symbol S(x, y, z) is interpreted as x + y = z; P (x, y, z) is meant to be x.y = z; L(x,y) is interpreted as x < y; and ≈ (x,y) is interpreted as x = y. For each of the sentences below, state whether it is true in Z, N, both or neither. Briefly(!) explain your answers in precise English.

(a) ∀x∃yP(x,y,x) (b) ∃y∀xP(x,y,x), (c) ∃x∀yS(x,y,y), (d) ∀x(L(x,0) → ∃yP(y,y,x)), (e) ∀x∀y(P(x,x,y)∧ ∼≈ (x,0) → L(0,x)), (f) ∀x∀y(P(x,x,y)∧ ∼≈ (x,0) → L(0,y)),

I have been completely stuck on this question for an hour with nothing but loose YouTube links and tears to solve for it, can someone please explain and answer this question?

Answer 1

Hint

You have to consider each formula in turn and try to "unpack" it into plain English; for (a) :

$∀x∃yP(x,y,x)$

it must be :

"for all number $x$, there is a number $y$ such that : $x = x \cdot y$" , i.e. $x$ is the product of $x$ and $y$.

The same for : $∃x∀yS(x,y,y)$, that reads as :

"there is a number $x$ such that, for all number $y$ : $y = x + y$" , i.e. $y$ is the sum of $x$ and $y$.

Having done this, you have to assess the truth-value of these statements when the variables $x$ and $y$ range over the sets $\mathbb Z$ of intergers and the set $\mathbb N$ of naturals respectively.

If you cannot "solve" them, you have to review the basic arithmetical properties ...