Textbook Problem

All but two of the following statements are correct ways to express the fact that a function f is onto. Find the two that are incorrect.

a. f is onto $\iff$ every element in its co-domain is the image of some element in its domain.

b. f is onto $\iff$ every element in its domain has a corresponding image in its co-domain.

c. f is onto $\iff \forall y\in Y,\exists x\in X$ such that $f\left(x\right)=y$ .

d. f is onto $\iff \forall x\in X,\exists x\in Y$ such that $f\left(x\right)=y$ .

e. f is onto $\iff$ the range of f is the same as the co-domain of f.