Question

Express the negation of these propositions using quantifiers, and then express the negation in English.

a) Some drivers do not obey the speed limit.

b) All Swedish movies are serious.

c) No one can keep a secret.

d) There is someone in this class who does not have a good attitude.

Short answer

a) The statement "Some drivers do not obey the speed limit” expressed using quantifiers would be $\mathbf{\neg }\mathbf{(}\exists \mathbf{x}\mathbf{\neg }\mathbf{P}\mathbf{(}\mathbf{x}\mathbf{\left)}\mathbf{\right)}\mathbf{\equiv }\mathbf{\forall }\mathbf{x}\mathbf{\neg }\mathbf{(}\mathbf{\neg }\mathbf{P}\mathbf{(}\mathbf{x}\mathbf{\left)}\mathbf{\right)}\mathbf{\equiv }\mathbf{\forall }\mathbf{xP}\mathbf{\left(}\mathbf{x}\mathbf{\right)}$and the negation in simple English can be expressed as “All drivers obey the speed limit ".

b) The statement” All Swedish movies are serious” expressed using quantifiers as$\mathbf{\neg }\mathbf{(}\mathbf{\forall }\mathbf{xQ}\mathbf{(}\mathbf{x}\mathbf{\left)}\mathbf{\right)}\mathbf{\equiv }\exists \mathbf{x}\mathbf{\neg }\mathbf{Q}\mathbf{\left(}\mathbf{x}\mathbf{\right)}$and the negation in simple English can be expressed as" There is a Swedish movie that is not serious ".

c)The statement “No one can keep a secret “expressed using quantifiers would be $\mathbf{\neg }\mathbf{(}\mathbf{\neg }\exists \mathbf{xR}\mathbf{(}\mathbf{x}\mathbf{\left)}\mathbf{\right)}\mathbf{\equiv }\exists \mathbf{xR}\mathbf{\left(}\mathbf{x}\mathbf{\right)}$and the negation in simple English can be expressed as " Somebody can keep a secret".

d) The statement "There is someone in this class who does not have a good attitude" expressed using quantifiers as $\mathbf{\neg }\mathbf{(}\mathbf{\neg }\exists \mathbf{x}\mathbf{\neg }\mathbf{S}\mathbf{(}\mathbf{x}\mathbf{\left)}\mathbf{\right)}\mathbf{\equiv }\mathbf{\forall }\mathbf{x}\mathbf{\neg }\mathbf{(}\mathbf{\neg }\mathbf{S}\mathbf{(}\mathbf{x}\mathbf{\left)}\mathbf{\right)}\mathbf{\equiv }\mathbf{\forall }\mathbf{xS}\mathbf{\left(}\mathbf{x}\mathbf{\right)}$and the negation in simple English can be expressed as “Everybody in this class has a good attitude ".

Step-by-step solution

INTERPRETATION SYMBOLS

Negation$\mathbf{\neg }\mathbf{p}$: not p

Conjunction$\mathbf{p}\mathbf{\wedge }\mathbf{q}\mathbf{:}\mathbf{p}$ and q

Existential quantification $\exists \mathbf{xP}\mathbf{\left(}\mathbf{x}\mathbf{\right)}$: There exists an element xin the domain such that P(x).

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

LOGICAL EQUIVALENCES

Double negation law:

$\mathbf{\neg }\mathbf{(}\mathbf{\neg }\mathbf{p}\mathbf{)}\mathbf{\equiv }\mathbf{p}$

De Morgan's laws:

$$\begin{gathered} \mathbf{\neg }\mathbf{(}\mathbf{p}\mathbf{\wedge }\mathbf{q}\mathbf{)}\mathbf{\equiv }\mathbf{\neg }\mathbf{p}\mathbf{\vee }\mathbf{\neg }\mathbf{q} \\ \mathbf{\neg }\mathbf{(}\mathbf{p}\mathbf{\vee }\mathbf{q}\mathbf{)}\mathbf{\equiv }\mathbf{\neg }\mathbf{p}\mathbf{\wedge }\mathbf{\neg }\mathbf{q} \end{gathered}$$

De Morgan's Laws for Qualifiers:

$$\begin{gathered} \mathbf{\neg }\exists \mathbf{xP}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{\equiv }\mathbf{\forall }\mathbf{x}\mathbf{\neg }\mathbf{P}\mathbf{\left(}\mathbf{x}\mathbf{\right)} \\ \mathbf{\neg }\mathbf{\forall }\mathbf{xP}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{\equiv }\exists \mathbf{x}\mathbf{\neg }\mathbf{P}\mathbf{\left(}\mathbf{x}\mathbf{\right)} \end{gathered}$$

Morgan’s Law

a) Let the domain be drivers and P(x)mean "x obeys the speed limit". We can then rewrite the given statement as:

$\exists \mathbf{x}\mathbf{\neg }\mathbf{P}\mathbf{\left(}\mathbf{x}\mathbf{\right)}$

The negation is then by De Morgan's Law for Qualifiers and the double negation law:

$\neg (\exists \mathbf{x}\mathbf{\neg }\mathbf{P}\mathbf{(}\mathbf{x}\mathbf{\left)}\mathbf{\right)}\mathbf{\equiv }\mathbf{\forall }\mathbf{x}\mathbf{\neg }\mathbf{(}\mathbf{\neg }\mathbf{P}\mathbf{(}\mathbf{x}\mathbf{\left)}\mathbf{\right)}\mathbf{\equiv }\mathbf{\forall }\mathbf{xP}\mathbf{\left(}\mathbf{x}\mathbf{\right)}$

This means: All drivers obey the speed limit.

b) Let the domain be Swedish movies and Q(x)mean "x is serious". We can then rewrite the given statement as:

$\mathbf{\forall }\mathbf{xQ}\mathbf{\left(}\mathbf{x}\mathbf{\right)}$

The negation is then by De Morgan's Law for Qualifiers:

$\mathbf{\neg }\mathbf{(}\mathbf{\forall }\mathbf{xQ}\mathbf{(}\mathbf{x}\mathbf{\left)}\mathbf{\right)}\mathbf{\equiv }\exists \mathbf{x}\mathbf{\neg }\mathbf{Q}\mathbf{\left(}\mathbf{x}\mathbf{\right)}$

This means: There is a Swedish movie that is not serious.

Double negation law

c) Let the domain be people and R(x) mean "x can keep a secret". We can then rewrite the given statement as "There is not a person that can keep a secret":

$\mathbf{\neg }\exists \mathbf{xR}\mathbf{\left(}\mathbf{x}\mathbf{\right)}$

The negation is then by the double negation law:

$\mathbf{\neg }\mathbf{(}\mathbf{\neg }\exists \mathbf{xR}\mathbf{(}\mathbf{x}\mathbf{\left)}\mathbf{\right)}\mathbf{\equiv }\exists \mathbf{xR}\mathbf{\left(}\mathbf{x}\mathbf{\right)}$

This means: Somebody can keep a secret.

De Morgan’s Law

d) Let the domain be people in this class and S(x) mean "x has a good attitude". We can then rewrite the given statement as:

$\exists \mathbf{x}\mathbf{\neg }\mathbf{S}\mathbf{\left(}\mathbf{x}\mathbf{\right)}$

The negation is then by De Morgan's Law for Qualifiers and the double negation law:

$\neg (\exists \mathbf{x}\mathbf{\neg }\mathbf{S}\mathbf{(}\mathbf{x}\mathbf{\left)}\mathbf{\right)}\mathbf{\equiv }\mathbf{\forall }\mathbf{x}\mathbf{\neg }\mathbf{(}\mathbf{\neg }\mathbf{S}\mathbf{(}\mathbf{x}\mathbf{\left)}\mathbf{\right)}\mathbf{\equiv }\mathbf{\forall }\mathbf{xS}\mathbf{\left(}\mathbf{x}\mathbf{\right)}$

This means: Everybody in this class has a good attitude.