1. **Problem:** Translate the given statements into propositional functions with quantifiers. 2. **Step 1: Define predicates** - Let $S(x)$ mean "$x$ is a senior." - Let $F(x)$ mean "$x$ is a freshman." - Let $J(x)$ mean "$x$ is a junior." - Let $D(x)$ mean "$x$ is dignified." - Let $P(x)$ mean "$x$ is pretty." - Let $G(x)$ mean "$x$ likes Greek." - Let $M(x)$ mean "$x$ likes Math." 3. **Step 2: Translate each statement** - (1) "All seniors are dignified": $$\forall x (S(x) \to D(x))$$ - (2) "No freshmen are dignified": $$\forall x (F(x) \to \neg D(x))$$ - (3) "Some seniors are not both pretty and dignified": $$\exists x (S(x) \wedge \neg (P(x) \wedge D(x)))$$ - (4) "Some seniors who like Greek are not pretty": $$\exists x (S(x) \wedge G(x) \wedge \neg P(x))$$ - (5) "No juniors like both Greek and Math": $$\forall x (J(x) \to \neg (G(x) \wedge M(x)))$$ 4. **Explanation:** - Universal quantifier $\forall$ means "for all." - Existential quantifier $\exists$ means "there exists at least one." - Implication $\to$ means "if... then..." - Negation $\neg$ means "not." - Conjunction $\wedge$ means "and." These translations capture the meaning of each English statement precisely using logical quantifiers and predicates. **Final answer:** 1. $$\forall x (S(x) \to D(x))$$ 2. $$\forall x (F(x) \to \neg D(x))$$ 3. $$\exists x (S(x) \wedge \neg (P(x) \wedge D(x)))$$ 4. $$\exists x (S(x) \wedge G(x) \wedge \neg P(x))$$ 5. $$\forall x (J(x) \to \neg (G(x) \wedge M(x)))$$