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) All dogs have fleas.b) There exists a horse that can add.c) Every koala can climb.d) No monkey can speak French.
e) There exists a pig that can swim and catch fish.

Answer 1

Answer:

a) All dogs have fleas.

X the set of all animals

D(y)=y is a dog

F(y)=y has fleas

∀X ( D(x) → F(x) )

¬∃X ( D(x) ^ F(x) )

b) There exists a horse that can add.

X the set of all animals

H(y)=y is a horse

A(y)=y can add

∃X ( H(x) ^ A(x) )

∀X ( H(x) → ¬A(x) )

c) Every koala can climb.

X the set of all animals

K(y)=y is a koala

C(y)=y can climb

∀X ( K(x) → C(x) )

¬∃X ( K(x) ^ C(x) )

d) No monkey can speak French.

X the set of all animals

M(y)=y is a Monkey

F(y)=y can speak French

∀X ( M(x) → ¬F(x) )

∃X ( M(x) ^ F(x) )

e) There exists a pig that can swim and catch fish.

X the set of all animals

P(y)=y is a pig

S(y)=y can swim

C(y)=y can catch fish

∃X ( P(x) ^ ( S(x) ^ C(x) )

∀X ( P(x) → ( ¬S(x) v ¬C(x) ) )