1. **Problem statement:** Express the given statements using predicates and quantifiers. 2. **Define predicates:** - Let $P(x)$ mean "$x$ is a passenger on an airline." - Let $M(x)$ mean "$x$ flies more than 30,000 miles in a year." - Let $F(x)$ mean "$x$ takes more than 25 flights during that year." - Let $E(x)$ mean "$x$ qualifies as an elite flyer." - Let $Man(x)$ mean "$x$ is a man." - Let $Woman(x)$ mean "$x$ is a woman." - Let $T(x)$ mean "$x$'s best previous marathon time in hours." - Let $Q(x)$ mean "$x$ qualifies for the marathon." 3. **Express statement (a):** "A passenger qualifies as an elite flyer if the passenger flies more than 30,000 miles in a year or takes more than 25 flights during that year." This can be written as: $$\forall x \big(P(x) \to (E(x) \leftrightarrow (M(x) \lor F(x)))\big)$$ 4. **Express statement (b):** "A man qualifies for the marathon if his best previous time is less than 4 hours and a woman qualifies if her best previous time is less than 5 hours." This can be written as: $$\forall x \big((Man(x) \land T(x) < 4) \to Q(x)\big) \quad \text{and} \quad \forall x \big((Woman(x) \land T(x) < 5) \to Q(x)\big)$$ **Final answers:** - (a) $$\forall x \big(P(x) \to (E(x) \leftrightarrow (M(x) \lor F(x)))\big)$$ - (b) $$\forall x \big((Man(x) \land T(x) < 4) \to Q(x)\big) \quad \text{and} \quad \forall x \big((Woman(x) \land T(x) < 5) \to Q(x)\big)$$