Function Definition Without a Single Word

[SteveL27]

Interesting approach. So there's one and only one universe that contains all possible mappings from sets A to B, and any f:A --> B is an element of this. I will research the topic further.

Well it's a set. You can get a handle on this by working out a simple example with small finite sets. For example let

A = {a1, a2, a3} and B = {b1, b2}. How many functions can there be from A to B?

Well a1 can go to b1 or b2.

a2 can go to b1 or b2.

a3 can go to b1 or b2.

So that gives us 2x2x2 = 8 possible functions from A to B. Each function is a set of exactly three ordered pairs, and there are only eight functions. Not too hard to write them all down.

Now you can see why we chose the notation B^A for the set of all functions from A to B. Because the set B^A has cardinality exactly |B|^|A|.