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.

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) ) )

256π units²   (answer b)

step-by-step explanation:

the formula for the surface area of a sphere of radius r

is             a = 4πr²

and so, if r = 8 units, then the surface area is:

a = 4π(8 units)² = 256π units²   (answer b)

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