Question

Translate the given statement into propositional logicexpressions:“A man qualifies for the marathon if his best previous time is less than 3 hours and a woman qualifies for the marathon if her best previous time is less than 3.5 hours.

Answer 1

The given statement can be interpreted in multiple ways. Take the below interpretation as an example.

Let’s define some propositions:

1. $M(x) = \text{“ Person }x\text{ is a man. ”}$
    
2. $W(x) = \text{“ Person }x\text{ is a woman. ”}$
    
3. $B(x) = \text{“ Previous best time of person }x\text{. ”}$
    
4. $Q(x) = \text{“ Person }x\text{ qualifies for the marathon. “}$
    

Statement:

“A man qualifies for the marathon if his best previous time is less than 3 hours and a woman qualifies for the marathon if her best previous time is less than 3.5 hours."

can be reinterpreted as:

“If a person is a man and his best previous time is less than 3 hours, then he qualifies for the marathon, and if a person is a woman, and her best previous time is less than 3.5 hours, then she qualifies for the marathon.”

$=\forall x[\underline{([M(x) \land (B(x)<3)] \rightarrow Q(x))} \land \underline{([W(x) \land (B(x)<3.5)] \rightarrow Q(x))}]$

$=\forall x[\{\underline{(M(x) \land (B(x)<3))} \lor \underline{(W(x) \land (B(x)<3.5))}\} \rightarrow Q(x)]$

which says: “A man with a previous best time less than 3 hours, or a woman with a previous best time of 3.5 hours, are qualified for the marathon.”