The problem is the following:
"Let $\varphi := \Big(P_1(x_1,x_2) \rightarrow P_1(x_2,x_1)\Big)$ be a formula in predicate logic. Describe (in as simple form as possible) exactly which values of $v(x_1)$ and $v(x_2)$ will make $\varphi$ hold, given the following structures:
(a) $<\mathbb{Z}; ;\text{"is equal to"}>$
(b) $<\mathbb{Z}; ; \text{"is less than"}>$. "
My reasoning goes as follows:
(a) Given this structure, we wish to find valuations $v(x_1), v(x_2)$ such that $$ v(x_1) = v(x_2) \Rightarrow v(x_2) = v(x_1). $$ Since $v(x_1), v(x_2) \in \mathbb{Z}$; we are simply looking for integers $a,b$ such that $$ a = b \Rightarrow b = a. $$ But this holds for any two identical integers $a,b$. Hence, $\varphi$ holds whenever $v(x_1) = v(x_2)$ in this structure.
(b) Here, we are instead looking for integers $a,b$ such that $$ a < b \Rightarrow b < a. $$ But this doesn't hold for any integers $a,b$. Hence, there are no valuations $v(x_1), v(x_2)$ for which $\varphi$ holds in this structure. $\ \square$
Is this correct? If it is, should I perhaps be more formal in how I present the answer? If not, am I misunderstanding something about how to deal with interpretations in predicate logic?