# A function is monotonically increasing if for each i, j ∈ ℤ m, i < j ⇒ f(i) ≤ f(j).

Heres the whole question but I'm completely lost on even how to start or what to do. If anyone wouldn't mind showing me how to do part a I think I might be able to nudge my way through the rest.

For this problem, we will define a special function f where m, n ∈ ℤ+:

$f : ℤ m → ℤ n$

A function is monotonically increasing if for each i, j ∈ ℤ m, i < jf(i)f(j).

(a) How many monotonically increasing functions are there with domain ℤ7 and codomain ℤ5?

(b) How many monotonically increasing functions are there with domain ℤ6 and codomain ℤ9?

(c) Generalize the results for parts (a) and (b) for arbitrary m , n ∈ ℤ+.

(d) How many monotonically increasing functions are there with domain ℤ10 and codomain ℤ8 where f (4) = 4?

(e) How many monotonically increasing functions are there with domain ℤ7 and codomain ℤ12 where f (5) = 9?

(f) Generalize the results for parts (d) and (e) for arbitrary m , n ∈ ℤ+.

• A question very like a) was asked $5$ hours ago by Hector. Maybe the answer will help you do the related questions. It will at least immediately let you deal with b) and c). – André Nicolas Nov 30 '15 at 4:23
• @AndréNicolas do you have the link? I can't find anything related to it :/ – RiGid Nov 30 '15 at 4:30
• Maybe this will work. – André Nicolas Nov 30 '15 at 4:41