I was solving questions of Discrete Mathematics and Its Applications
By Kenneth H. Rosen Chapter 1 The foundations: Logic and Proofs , when i got stuck at this problem; two similar problems are having different expressions with predicates and quantifiers
35. Express each of these statements using predicates
and quantifiersb) 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.Solution: $∀x(((M(x)∧T (x, 3))∨(¬M(x) ∧ T (x, 3.5)))→ Q(x))$,
where $Q(x)$ is “x qualifies for the marathon,” $M(x)$ is “x is a man,” and $T(x, y)$ is “x has run the marathon in less than y hours”c) A student must take at least 60 course hours, or at least 45 course hours and write a master's thesis, and receive a grade no lower than a B in all required courses, to receive a master's degree.
Solution: $M →((H (60) ∨ (H(45) ∧ T )) ∧ ∀y G(B, y))$, where $M$ is the
proposition “The student received a masters degree,” $H(x)$ is
“The student took at least $x$ course hours,” $T$ is the proposition
“The student wrote a thesis,” and $G(x, y)$ is “The person got
grade $x$ or higher in course $y$”.
Please help me understand why in first example "if his best previous time is less
than 3 hours(for man) or if her best previous time is less
than 3.5 hours (for woman)" forms the hypothesis ie p of conditional statement p→ q and in second example "A student must take at least 60 course hours, or
at least 45 course hours and write a master's thesis,
and receive a grade no lower than a B in all
required courses" is not taken as hypothesis p.
Best Answer
In general, we translate a statement "$B$ if $A$" or, equivalently, "if $A$, then $B$" as: $$A\to B$$ which means that it is never the case that $A$ and $\neg B$.
Now, your second statement "A student must [conditions] to receive a master's degree" is equivalent to "A student receives a master's degree only if [conditions]", meaning that it is never the case that a student will receive a master's degree and not satisfy [conditions]. Therefore: $$\text{master}\to\text{conditions}$$ If we'd written $\text{conditions}\to\text{master}$ instead, then a student could receive a master's degree while not satisfying [conditions], hence breaking the only if requirement.