GATE CS 2005 Q41 — Propositional and First Order Logic

To translate the English sentence "Every teacher is liked by some student" into first-order predicate logic, we can break it down step-by-step:

1. "Every teacher $x$ ..."
   This indicates a universal quantification over all individuals $x$. If $x$ is a teacher, then some property holds: $$\forall(x) [\text{teacher}(x) \rightarrow \dots]$$ Note: We use the implication $(\rightarrow)$ with the universal quantifier $(\forall)$ because we only care about individuals who are teachers.
2. "... is liked by some student $y$"
   For each teacher $x$, there exists at least one individual $y$ such that $y$ is a student and $y$ likes $x$: $$\exists(y) [\text{student}(y) \land \text{likes}(y, x)]$$ Note: We use the conjunction $(\land)$ with the existential quantifier $(\exists)$ because we are asserting the simultaneous existence of someone who is both a student and likes $x$. Using an implication $(\rightarrow)$ here would be vacuously true if there exists any non-student in the domain, which is incorrect.

Combining these two parts, we get:

This matches Option B.