I have the task:
Consider the formula $\forall x (Q(x,b) \to Q(b,x))$ where $Q$ is a binary predicate symbol and $b$ is an individual constant. State
i) one non-satisfying, and
ii) one satisfying
FOL structure (with justification).
Progress so far:
First, define what satisfiability means in relation to a formula. Since the formula being considered is of the conditional form, a structure $\mathfrak{M}$ needs to be found such $Q(x,b)$ is not satisfied or $Q(b,x)$ is satisfied or $Q(x,b)$ and $Q(b,x)$ are satisfied.
Am I on the right track? If so, what can I do as a next step?
Your attempt is correct for question ii), not for question i).
i) You should find a structure $\mathfrak{M}$ such that $\mathfrak{M} \models \lnot \forall x (Q(x,b) \to Q(b,x))$, i.e. $\mathfrak{M} \models \exists x (Q(x,b) \land \lnot Q(b,x))$. It means that in $\mathfrak{M}$ you should have an element $x$ such that $Q(x,b)$ is satisfied and $Q(b,x)$ is not satisfied. Now, let $\mathfrak{M}$ be the structure whose domain is $\{0,1\}$ and such that $b$ is interpreted by $1$ and $Q(x,y)$ is interpreted as $x < y$: then, $Q(0,b)$ but $\lnot Q(b,0)$ in $\mathfrak{M}$, hence $\mathfrak{M} \models \exists x (Q(x,b) \land \lnot Q(b,x))$ taking $x = 0$ as witness.
ii) The formula $\forall x (Q(x,b) \to Q(b,x))$ is satisfied by any structure $\mathfrak{M}$ where a symmetric relation $Q$ can be defined. For instance...