Discrete Math a Let f AB be an injective function on sets A

(Discrete Math)

a) Let f: AB be an injective function on sets A Z and B Z. Prove that there exists some surjective function g: BA.

b) Let f: AB be a surjective function on sets A Z and B Z. Prove that there exists some injective function g: BA.

Solution

sol) yes there exists such functions.

since f:A-->B is injective function which implies that every element of set A has pre image in set B .since if we take its inverse function g:B-->A implies that cardinality of B should be equal to cardinality of A.otherwise its inverse does not exixt.which shows that there exist surjective function .