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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!"

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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 $|C|^{(|A||B|)}$

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  • $\begingroup$ Brilliant. Thanks a lot. Exactly what I was looking for. $\endgroup$
    – Harrison Nguyen
    Sep 18, 2013 at 6:11
  • $\begingroup$ 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. $\endgroup$
    – Ross Millikan
    Sep 18, 2013 at 13:01

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