Question

A discrete mathematics class contains 1 mathematics major who is a freshman, 12 mathematics majors who are sophomores, 15 computer science majors who are sophomores, 2 mathematics majors who are juniors, 2 computer science majors who are juniors, and 1 computer science major who is a senior. Express each of these statements in terms of quantifiers and then determine its truth value.

a) There is a student in the class who is a junior.

b) Every student in the class is a computer science major.

c) There is a student in the class who is neither a mathematics major nor a junior.

d) Every student in the class is either a sophomore or a computer science major.

e) There is a major such that there is a student in the class in every year of study with that major.

Short answer

$\left(\mathbf{a}\right)\boxed{\exists s\exists mP(s,junior,m)}$, the statement is true.

$$\begin{gathered} \left(\mathbf{b}\right)\boxed{\forall \mathrm{s}\exists \mathrm{c}\exists \mathrm{mP}(\mathrm{s},\mathrm{c},\mathrm{Computer}\mathrm{science})},\mathrm{the}\mathrm{statement}\mathrm{is}\mathrm{false}. \\ \left(\mathbf{c}\right)\boxed{\exists \mathrm{s}\exists \mathrm{c}\exists \mathrm{m}\left(\mathrm{P}\right(\mathrm{s},\mathrm{c},\mathrm{m})\wedge (\mathrm{c}\ne \mathrm{junior})\wedge (\mathrm{m}\ne \mathrm{mathematics}\left)\right)},\mathrm{the}\mathrm{statement}\mathrm{is}\mathrm{true}. \\ \left(\mathbf{d}\right)\boxed{\forall (\exists \mathrm{cP}(\mathrm{s},\mathrm{c},\mathrm{computer}\mathrm{Science})\vee \exists \mathrm{mP}(\mathrm{s},\mathrm{sophom}\mathrm{ore},\mathrm{m})},\mathrm{the}\mathrm{statement}\mathrm{is}\mathrm{false}. \\ \mathbf{\left(}\mathbf{e}\mathbf{\right)}\boxed{\exists \mathrm{m}\forall \mathrm{c}\exists \mathrm{sP}(\mathrm{s},\mathrm{c},\mathrm{m})},\mathrm{the}\mathrm{statement}\mathrm{is}\mathrm{false}. \end{gathered}$$

Step-by-step solution

Given

For the data provided express these statements, usingquantifiers and find the truth value

(a) Determine the student in the class who is a junior

There is a student in the class who is a junior

Fromdata provided:

Let P(s,c,m) be the statement that student is s who has a class c and with major in m .

s:Range over the student in the class

m: Range over all the majors that are possible

c: Range over a total class standing of four.

Thus, the required preposition here is

$\boxed{\exists s\exists mP(s,junior,m)}$

From the data provided it can be said that this statement is true.

(b) Determine the student in the class is a computer science major

Every student in the class is a computer science major.

Thus, the required preposition here is

$\boxed{\forall \mathrm{s}\exists \mathrm{c}\exists \mathrm{mP}(\mathrm{s},\mathrm{c},\mathrm{Computer}\mathrm{science})}$

From the data provided it can be seen that There are some students who are mathematics major also.

Thus, this statement is false.

(c) Determine the student in the class who is neither a mathematic  major nor a junior

There is a student in the class who is neither a mathematic major nor a junior.

Thus, the required preposition here is

$\boxed{\exists \mathrm{s}\exists \mathrm{c}\exists \mathrm{m}\left(\mathrm{P}\right(\mathrm{s},\mathrm{c},\mathrm{m})\wedge (\mathrm{c}\ne \mathrm{junior})\wedge (\mathrm{m}\ne \mathrm{mathematics}\left)\right)}$

From the data provided as there is a sophomore who has a major as computer,

this statement is true.

(d) Determine student in the class is either a sophomore or a  computer science major

Every student in the class is either a sophomore or a computer science major.

Thus, the required preposition here is

$\boxed{\forall (\exists \mathrm{cP}(\mathrm{s},\mathrm{c},\mathrm{computer}\mathrm{Science})\vee \exists \mathrm{mP}(\mathrm{s},\mathrm{sophom}\mathrm{ore},\mathrm{m})}$

From the data provided as there is a fresh man who is a major in mathematics,

Thus this statement is false.

(e) Determine the major such that there is a student in the class in every year of study with that major

There is a major such that there is a student in the class in every year of study with that major.

The required preposition here is

$\exists m\forall c\exists sP(s,c,m)$.

It can be seen that there is no senior who has a major in mathematics, and there is no computer students who are freshman major, that m cannot be either computer or mathematics. it cannot be any other major also.

Thus this statement is false.

Thus, the required result is found.