If |A|<|B| there are NO surjections from A to B and permut(|B|,|A|) injections.
The number of mapping from A to B is
(a) Suppose that the set A has exactly two elements and the set B has exactly three. You have to construct explicitly all functions from A to B, and to decide whether each of them is injective and surjective. There are nine such functions – let us label them (arbitrarily) as f1, f2, . . ., f9.
Let A = {1, 2}, and B = {a, b, c}. In the table below fill out all possible values of f1,
f2, . . ., f9 on 1 and 2.
Function fi(1) fi(2) Injective? Surjective?
f1 a a No No
f2 a b Yes No
f3 a c Yes No
f4 ? ? ? ?
f5 ? ? ? ?
f6 ? ? ? ?
f7 ? ? ? ?
f8 ? ? ? ?
f9 ? ? ? ?
(b) Now let A has exactly three elements and B has exactly two: A = {1, 2, 3}, B = {a, b}.
Again, construct explicitly all functions from A to B and determine whether they are injective and surjective. Put your results in a table similar to the table in part (a).
(c) Let the set A have exactly m elements and B have exactly n (where n,m 2 N). How many different functions are there from A to B?