I have the sentence $\phi = \forall _x\exists_y\exists_z(p(x,y)\land p(z,y)\land(p(x,z)\rightarrow p(z,x))),$ and the following interpretations, and the task is to evaluate $\phi$'s true value in each:
a) The interpretation $M$ has as its domain the set of natural numbers, and $ p^M = \{(m,n)| m < n\} $
b)The interpretation $M'$ has as its domain the set of natural numbers, and $ p^{M'} =\{(m,2m)|\;m\;is\;a\;natural\;number\;\} $
c)The interpretation $M''$ has as its domain the set of natural numbers, and $ p^{M''} =\{(m,n)|m < n+1\}.$
I have provided some proofs, however I don't know if they are correct and if they are sufficiently formalized:
a)
b)
c)
Edit
a) For any x, take y=x+1 and z=x, and that will make everything true.
b) For any x, take y=2x and z=x, then everything will be true.
c) For any x, take y=x and z=x, then everything will be true.
It makes little sense to assign a truth-value to a formula like $p(x,y)$, because in this formula, $x$ and $y$ are free variables, and you only obtain a sentence once those variables get quantified.
In fact, it crucially depends on how the variables are quantified. For example, if we take the first interpretation $M$, then a sentence like:
$$\forall x \forall y \ p(x,y)$$
comes to mean:
"Every natural number is smaller than all natural numbers"
which is clearly False.
But if you use existentials, you get:
$$\exists x \exists y \ p(x,y)$$
which would mean "some number is smaller than some natural number"
which is True.
And, if you use a mix of quantifiers, then:
$$\forall x \exists y \ p(x,y)$$
means: "Every natural number is smaller than some natural number", which is True (indeed, for every natural number $xZ$, I can pick some natural $y$ (e.g. we can pick $y=x+1$) such that $x<y$
but:
$$\exists x \forall y \ p(x,y)$$
would mean: "There is some natural number that is smaller than all natural numbers"
which is False: even if you pick $0$, that is still not smaller than all natural numbers, since $0$ is not smaller than itself (if you have the natural numbers start with $1$, then we can point out that $1$ is not smaller than $1$)
So, long story short: you need to find the value of the whole sentence $\phi = \forall _x\exists_y\exists_z(p(x,y)\land p(z,y)\land(p(x,z)\rightarrow p(z,x)))$ given the 3 different interpretations.