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# Let D be the set of all students at your school, and let M(s) be a ”s is a math major”, let C(s)”s is a computer science student”, and let E(s) be ”s is an engineering student.” Express each of the following statements using quantifiers, variables, and predicates M(s), C(s) and E(s)

Question
Let D be the set of all students at your school, and let M(s) be a ”s is a math major”, let C(s)”s is a computer science student”, and let E(s) be ”s is an engineering student.” Express each of the following statements using quantifiers, variables, and predicates M(s), C(s) and E(s)

## Answers (1)

2021-01-05
Solution:
$$\displaystyle{D}=$$ set of all students at your school (domain)
$$\displaystyle{M}{\left({s}\right)}=$$ ”s is a math major”
$$\displaystyle{C}{\left({s}\right)}=$$ ”s is a computer science student”
$$\displaystyle{E}{\left({s}\right)}=$$ "s is an engineering student”
1)We can rewrite the given statement as: ”There is at least one student s $$\displaystyle\in{D}$$ such that s is a math major and s is an engineering student”
There is at least one” implies an existential statement, thus we need to use $$\displaystyle\exists$$ instead.
Replace ”s is a math major” by $$\displaystyle{M}{\left({s}\right)}$$, ”s is an engineering student” by $$\displaystyle{E}{\left({s}\right)}$$ and ”and” by $$\displaystyle\wedge.$$
existss $$\displaystyle\in{D}$$ such that $$\displaystyle{M}{\left({s}\right)}\wedge{E}{\left({s}\right)}$$
2) We can rewrite the given statement as: ”For every student s $$\displaystyle\in{D},$$ if s is a computer science student, then s is an engineering student.”
”For every” implies an universal statement, thus we need to use $$\displaystyle\forall$$ instead.
Replace ”s is a computer science student” by $$\displaystyle{C}{\left({s}\right)},$$ ”s is an engineering student” by $$\displaystyle{E}{\left({s}\right)}$$ and ”if-then” by $$\displaystyle\rightarrow$$. $$\displaystyle\forall{s}\in{D},{C}{\left({s}\right)}\rightarrow{E}{\left({s}\right)}$$
3) We can rewrite the given statement as: ”For every student s $$\displaystyle\in{D},$$ if s is a computer science student, then s is not an engineering student.” ”For every” implies an universal statement, thus we need to use \forall instead. Replace ”s is a computer science student” by $$\displaystyle{C}{\left({s}\right)},$$ ”s is an engineering student” by (s), "ifthen” by $$\displaystyle\rightarrow$$ and ”not” by ~ $$\displaystyle\forall{s}\in{D},{C}{\left({s}\right)}\rightarrow\sim{E}{\left({s}\right)}$$
4) We can rewrite the given statement as: ”There is at least one student s $$\displaystyle\in{D}$$ such that s is a math major and s is a computer science student”. ”There is at least one” implies an existential statement, thus we need to use \exists instead. Replace ”s is a math major” by $$\displaystyle{M}{\left({s}\right)},$$ ”s is a computer science student” by
PSKC(s) and ”and” by $$\displaystyle\wedge.$$
$$\displaystyle\exists{s}\in{D}$$ such that $$\displaystyle{M}{\left({s}\right)}\wedge{C}{\left({s}\right)}$$
5) We can rewrite the given statement as: ”There is at least one student s $$\displaystyle\in{D}$$ such that s is a computer science student and s is an engineering student, and There is at least one student $$\displaystyle{t}\in{D}$$ such that t is a computer science student and t is an engineering student” ”There is at least one” implies an existential statement, thus we need to use \exists instead.
Replace ”s is a computer science student” by $$\displaystyle{C}{\left({s}\right)},$$ ”s is an engineering student” by $$\displaystyle{E}{\left({s}\right)},$$ and” by $$\displaystyle\wedge$$ and ”not” by ~ $$\displaystyle{\left(\exists{s}\in{D}\right.}$$ such that $$\displaystyle{C}{\left({s}\right)}\wedge{E}{\left({s}\right)}{)}\wedge$$ ($$\displaystyle\exists{t}\in{D}$$ such that $$\displaystyle{C}{\left({s}\right)}\wedge\sim{E}{\left({s}\right)}$$)

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