Answer
a) $\neg (\exists x A(x))$
b) $\neg \forall x A(x)$
c)$\forall x(B(x)\rightarrow A(x))$
d) $\exists x(A(x) \land B(x))$
e) $\forall x(A(x) \land B(x))$
f)$(\neg \forall x B(x))\lor (\exists x \neg A(x))$
Work Step by Step
a) Let the domain be all people in the world and A(x) means "x is perfect".
" No one" means that there does not exist a person ( for which the statement holds)
$$\neg (\exists x A(x))$$
b) Let the domain be all the people in the world and A(x) means " x is perfect".
"Not everyone" means not all people, thus
$$\neg \forall x A(x)$$
c) Let the domain be all people in the world, A(x) means "x is perfect" and B(x) means "x is your friend".
"All" means everybody and thus
If x is your friend, then x has to be perfect.
$$\forall x(B(x)\rightarrow A(x))$$
d)Let the domain be all people in the world, A(x) means "x is perfect" and B(x) means "x is your friend".
"At least one" means that there exists a person $\exists x$ . This person x also has to be your friend and has to be perfect.
$$\exists x(A(x) \land B(x))$$
e)Let the domain be all the people in the world, A(x) means " x is perfect" and B(x) means " x is your friend".
"Everyone" means $\forall$x. These people x have to be your friend and have to be perfect.
$$\forall x(A(x) \land B(x))$$
f) Let the domain be all people in the world, A(x) means "x is perfect" and B(x) means " x is your friend".
"Not everyone" means not for every person $\neg \exists x$. "Someone" means that there exists a person $\exists x$.
$$(\neg \forall x B(x))\lor (\exists x \neg A(x))$$
