Question

 Express each of these statements using quantifiers. Then form the negation of the statement, so that no negation is to the left of a quantifier. Next, express the negation in simple English. (Do not simply use the phrase “It is not the case that.”) a) Some old dogs can learn new tricks. b) No rabbit knows calculus. c) Every bird can fly. d) There is no dog that can talk. e) There is no one in this class who knows French and Russian.

Answer 1

Solution:

1. Origin Statement: "Some dogs can learn new tricks."

In symbols: $\exist x\in Dogs\, \, Can(x,LearnTricks)$

Negation: $\forall x\in Dogs\, \, \neg Can(x,LearnTricks)$

in English: "No dog can learn new tricks."

2. Origin Statement: "No rabbit knows calculus."

In symbols: $\forall x\in Rabbits\, \neg Knows(x,Calculus)$

Negation: $\exist x\in Rabbits\, Knows(x,Calculus)$

in English: "Some rabbits know calculus."

3. Origin Statement: "Every bird can fly."

In symbols: $\forall x\in Birds\,\, Can(x,Fly)$

Negation: $\exist x\in Birds\, \neg Can(x,Fly)$

in English: "Some birds can not fly."

4. Origin Statement: "There is no dog that can talk."

In symbols: $\neg(\exist x\in Dogs\, Can(x,Talk))$ or $\forall x\in Dogs\, \neg Can(x,Talk)$

Negation: $\exist x\in Dogs\, Can(x,Talk)$

in English: "There exists dog that can talk."

5. Origin Statement: "There is no one in this tutorial who knows French and Russian."

In symbols: $\neg(\exist x\in Tutorial\, Knows(x,French)\wedge Knows(x,Russian))$ or $\forall x\in Tutorial\, \neg Knows(x,French)\vee \neg Knows(x,Russian)$

Negation: $\exist x\in Tutorial\, Knows(x,French)\wedge Knows(x,Russian)$

in English: "Someone in this tutorial knows French and Russian."