counter-example: and defined by then but

: as an exercise prove that (B) is true if and only if is one-to-one!

counter-example: and defined by then

(C) for every

if then for some thus i.e. this is true for any function now if is onto and then for some

(D) for every if and only if

hence which means and so therefore conversely, if for any then choosing we

have and thus i.e. is onto.