Question

Express each of these statements using predicates and quantifiers.

a) A passenger on an airline qualifies as an elite flyer if the passenger flies more than 25,000 miles in a year or takes more than 25 flights during that year.

b) 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.

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.

d) There is a student who has taken more than 21 credit hours in a semester and received all A’s.

Short answer

$$\begin{gathered} \mathrm{a})\mathbf{\forall }\mathbf{x}\mathbf{(}\mathbf{\left(}\mathbf{Q}\mathbf{\right(}\mathbf{x}\mathbf{)}\mathbf{\vee }\mathbf{R}\mathbf{(}\mathbf{x}\mathbf{\left)}\mathbf{\right)}\mathbf{\to }\mathbf{P}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{\left)} \\ \mathrm{b}\right)\mathbf{}\mathbf{\forall }\mathbf{x}\mathbf{\left[}\mathbf{\right(}\mathbf{A}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{\wedge }\mathbf{S}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{)}\mathbf{\to }\mathbf{U}\mathbf{(}\mathbf{x}\mathbf{\left)}\mathbf{\right]}\mathbf{\wedge }\mathbf{\forall }\mathbf{x}\mathbf{\left[}\mathbf{\right(}\mathbf{B}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{\wedge }\mathbf{T}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{)}\mathbf{\to }\mathbf{U}\mathbf{(}\mathbf{x}\mathbf{\left)}\mathbf{\right]} \\ \mathrm{c})\mathbf{\forall }\mathbf{x}\mathbf{[}\mathbf{\left[}\mathbf{\right(}\mathbf{C}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{60}\mathbf{)}\mathbf{\vee }\mathbf{\left(}\mathbf{C}\mathbf{\right(}\mathbf{x}\mathbf{,}\mathbf{45}\mathbf{)}\mathbf{\wedge }\mathbf{D}\mathbf{(}\mathbf{x}\mathbf{\left)}\mathbf{\right)}\mathbf{)}\mathbf{\wedge }\mathbf{E}\mathbf{(}\mathbf{x}\mathbf{\left)}\mathbf{\right]}\mathbf{\to }\mathbf{F}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{\left]} \\ \mathrm{d}\right)\mathbf{}\mathbf{\exists }\mathbf{x}\mathbf{\left(}\mathbf{G}\mathbf{\right(}\mathbf{x}\mathbf{)}\mathbf{\wedge }\mathbf{H}\mathbf{(}\mathbf{x}\mathbf{\left)}\mathbf{\right)} \end{gathered}$$

Step-by-step solution

∀x((Q(x)∨R(x))→P(x))

Let the domain be all passengers on the airline, P(x) mean x" is an elite flyer", Q(x) mean " x flies more than25,000 miles" andR(x) mean "xtakes more than 25 flights during that year".

∀x[(A(x)∧S(x))→U(x)]∧∀x[(B(x)∧T(x))→U(x)]

Let A(x) be "x is a man" and B(x) be " is a woman".$\mathbf{T}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{)}\mathbf{\to }\mathbf{U}\mathbf{(}\mathbf{x}\mathbf{\left)}\mathbf{\right]}$Let S(x) be "xhas a best previous time less than 3hours" and T(x) be "x has a best previous time less than 3.5 hours”. Let U(x) be " x qualifies for the marathon".

∀x[[(C(x,60)∨(C(x,45)∧D(x)))∧E(x)]→F(x)]

LetC(x,y)be”x takes at least yhours",D(x) be” x.

$\mathbf{E}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{]}\mathbf{\to }\mathbf{F}\mathbf{(}\mathbf{x}\mathbf{\left)}\mathbf{\right]}$Writes a master's thesis" and E(x)be” x receives a grade no lower than B in all required courses" and F(x) be” x receives a master's degree".

∃x(G(x)∧H(x))

Let G(x) be” x has taken more than 21 credit hours in a semester" and H(x) be” x received all A's".