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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