 Solution

Extend the first-order variable assignment recursively to first-order terms byFor atomic formulas, $\mathcal{A}\models t=s[v]$ exactly when $\tilde{v}(t)=\tilde{v}(s)$. Extend to the Boolean connectives by their truth tables, for example implication is false exactly when its antecedent is true and its consequent false. Quantifiers use modified first-order variable assignments:This recursion defines satisfaction for a set structure for every first-order formula.