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 \(\displaystyle{C}{\left({s}\right)}\wedge\sim{E}{\left({s}\right)}\))

\(\displaystyle{D}=\) 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 \(\displaystyle{C}{\left({s}\right)}\wedge\sim{E}{\left({s}\right)}\))