# Sets, Total Functions, Equality

Suppose you have finite Sets $$A$$, $$B$$, $$C$$.

The function from $$X \to_\text{total}Y$$ represents a set of all of the total functions from set $$X$$ to set $$Y$$.

Ex:

Suppose $$X$$ is the set $$\{1,0\}$$, $$Y$$ is the set $$\{3,4\}$$

$$X \to_\text{total}Y$$ would be the set $$\{(1,3)(0,3);(1,4)(0,4);(1,3)(0,4);(1,4)(0,3)\}$$ which has four elements in it.

Now, the number of elements in $$A\to_\text{total}(B\to_\text{total}$$C)

will always equal the number of elements in $$B\to_\text{total}(A\to_\text{total}$$C).

(At least I think so, I tried with a few different sets by hand).

What is this property called?

It reminds me of the multiplicative property of equality but I don't think that's what it would be formally called, or if it is a property at all. I'm mainly looking for a better way to describe it than "I tried it on a bunch of different sets and it worked!"

It represents one of the laws of exponents, that $(a^b)^c=a^{bc}$. The number of total functions from $X$ to $Y$ is $|Y|^{|X|}$ where the bars represent the cardinality of the set. Then the first of your expressions is $\left( |C|^{|B|}\right)^{|A|}$ and the other is $\left( |C|^{|A|}\right)^{|B|}$. Both equal $|C|^{(|A||B|)}$
• I realized later that you can think of them as two different ways of looking at the set of functions from $(A \times B) \to C$. One picks the first coordinate (from A) first, the other picks the second coordinate first. – Ross Millikan Sep 18 '13 at 13:01