Total number of functions = m^n (imagine distributing n objects in m boxes, and for each object there are m options, notice that you are distributing all n objects and all m boxes in B may not be used)
One-One function (Injective): Easiest of the lot, for every n, there exists one and only one m in B. One-one function exists only when m>=n. Basically, each box can hold only 1 object.
Onto function (Surjective): In this case, all the m boxes must get at least 1 object. So, it is obvious that the number of objects must exceed the number of boxes i.e. n>=m. Only then, we will have an onto function. You can find the inclusion exclusion formula to calculate this, but I prefer the permutation and combination method.
One-one onto (Bijective): Combination of the two above. You need to use all the boxes and put only one object per box. So, in this m = n.
PS: For diagrams, refer wikipedia. Also, if there are mistakes in this, do notify, I am not an expert in this :splat:
@Estallar12 please confirm and provide some questions on this concept.