I'm learning about the satisfiability of consistent sets and I'm having trouble approaching an exercise.
Let $S := \{R\}$ with unary $R$ and let $\phi := \{\exists x\,Rx\} \cup\{\neg Ry \mid y\text{ is a variable}\}$
Show that $\phi$ is satisfiable and therefore consistent.
I'm not really sure how to approach this proof. Is there a proper procedure as to proving satisfiability of a set of sentences?
Because I am lazy, I tend to not define "satisfiability" for formulas, only for sentences. This means I get to ignore free variables, and not consider what they represent.
Anyway, you can prove satisfiability directly: By exhibiting a model, (augmented with an interpretation for the variables). We need an object in the extension of $R$ to satisfy the first sentence, and, we need some number of objects not in the extension of $R$ to satisfy the formulas, so our model $\mathcal{M}$ should have at least two objects $\{0,1\}$, lets try $R^{\mathcal{M}} = \{0\}$. Now I need an intepretation of the variables, as I don't want them to be in the extension of $R$, map each variable to $1$.