Answer
a) $\forall x (P(x) $$\rightarrow$ $\neg$ $ Q(x))$
b) $\forall x (R(x) $$\rightarrow$ $\neg$ $ S(x))$
c) $\forall x ($$\neg Q(x) $ $\rightarrow$ $ S(x))$
d) $\forall x (P(x) $$\rightarrow$ $\neg$ $ R(x))$
e) Yes
Work Step by Step
a) Rewrite the given statement " Babies are illogical" in If-then form.
If x is a baby, then x is not logical. This statement if true for all x.
$\forall x (P(x) $$\rightarrow$ $\neg$ $ Q(x))$
b) Rewrite the given statement " Nobody is despised who can manage a crocodile" in If-then form.
If x is able to manage a crocodile, then x can't be despised.This statement if true for all x.
$\forall x (R(x) $$\rightarrow$ $\neg$ $ S(x))$
c)Rewrite the given statement " Illogical persons are despised" in If-then form.
If x is not logical, then x is despised.This statement if true for all x.
$\forall x ($$\neg Q(x) $ $\rightarrow$ $ S(x))$
d)Rewrite the given statement " Babies cannot manage crocodile" in If-then form.
If x is baby, then x cannot manage a crocodile.This statement if true for all x.
$\forall x (P(x) $$\rightarrow$ $\neg$ $ R(x))$
e) Yes.
Suppose x is baby.
Then from a) x is illogical.
Then from c) x is despised.
But the second premise says that if x could manage a crocodile, then x would not be despised. Therefore x cannot manage a crocodile. Thus we have proved that babies cannot manage crocodiles.
