Question

Let P(x) be the statement “Student x knows calculus” and let Q(y) be the statement “Class y contains a student who knows calculus.” Express each of these as quantifications of P(x) and Q(y).

a) Some students know calculus.

b) Not every student knows calculus.

c) Every class has a student in it who knows calculus.

d) Every student in every class knows calculus.

e) There is at least one class with no students who know calculus.

Short answer

(a) \(\exists xP\left( x \right)\)

(b)\(\neg \forall xP\left( x \right)\)

(c) \(\forall yQ\left( y \right)\)

(d) \(\left( {\forall xP\left( x \right)} \right) \wedge \left( {\forall yQ\left( y \right)} \right)\)

(e) \(\exists y\neg Q\left( y \right)\)

Step-by-step solution

Interpretation of symbols

Negation\(\neg p\): “not p.”

Disjunction\(p \vee q\): p or q

Conjunction\(p \wedge q\): p and q

Exclusive or\(p \oplus q\): p or q, but not both

Conditional statement\(p \to q\): if p, then q

Bi-conditional statement\(p \leftrightarrow q\): p, if and only if q

Existential quantification\(\exists xP\left( x \right)\): There exist an element x in the domain such that P(x).

Universal quantification\(\forall xP\left( x \right)\): P(x) for all values of x in the domain.

Express the given statement as quantifications of P(x) and Q(y) (a)

It is given that,

\(\begin{array}{c}P\left( x \right) = {\rm{Student}}\;x\;{\rm{knows}}\;{\rm{calculus}}\\Q\left( y \right) = {\rm{Class}}\;y\;{\rm{contains}}\;{\rm{a}}\;{\rm{student}}\;{\rm{who}}\;{\rm{knows}}\;{\rm{calculus}}\end{array}\)

Rewrite the given statement as “There exist a student who knows calculus.”

Use the interpretation, to rewrite the statement as mathematical expression.

\(\exists xP\left( x \right)\)

Therefore, the required quantifications is \(\exists xP\left( x \right)\).

Express the given statement as quantifications of P(x) and Q(y)(b)

Rewrite the given statement as “Not all students know calculus.”

Use the interpretation, to rewrite the statement as mathematical expression.

\(\neg \forall xP\left( x \right)\)

Therefore, the required quantificationsis \(\neg \forall xP\left( x \right)\).

Express the given statement as quantifications of P(x) and Q(y) (c)

Rewrite the given statement as “All classes contain a student who knows calculus.”

Use the interpretation, to rewrite the statement as mathematical expression.

\(\forall yQ\left( y \right)\)

Therefore, the required quantificationsare\(\forall yQ\left( y \right)\).

Express the given statement as quantifications of P(x) and Q(y) (d)

Rewrite the given statement as “All students know calculus and every class contains a student who knows calculus.”

Use the interpretation, to rewrite the statement as mathematical expression.

\(\left( {\forall xP\left( x \right)} \right) \wedge \left( {\forall yQ\left( y \right)} \right)\)

Therefore, the required quantificationsare\(\left( {\forall xP\left( x \right)} \right) \wedge \left( {\forall yQ\left( y \right)} \right)\).

Express the given statement as quantifications of P(x) and Q(y) (e)

Rewrite the given statement as “There exists a class that does not contain a student who knows calculus.”

Use the interpretation, to rewrite the statement as mathematical expression.

\(\exists y\neg Q\left( y \right)\)

Therefore, the required quantificationsare\(\exists y\neg Q\left( y \right)\).