"$((A\times B) \to C)$" denotes what?

Question

I'm having some trouble understanding notation. The question is

For any three sets $A,B,C$ ,
$((A\times B) \to C) =_c (A \to (B\to C))$

Exactly what does "$((A\times B) \to C)$" denote? Is it the set of all mappings from $A\times B$ to $C$? Or just a mapping?

I found this in Yiannis Moschovakis "Notes on set Theory".

Answer 1

Usually, the symbols $A \times B \to C$ denote a specific mapping from the Cartesian product of $A$ and $B$ to $C$. It could be that the author means that setting parentheses around it should denote the set of all mappings $A \times B \to C$. That is, in more standard notation $(A \times B \to C) = \hom(A\times B, C)$.

It is a standard fact that $\hom(A \times B, C) = \hom(A, \hom(B,C)$ (can you see why?)

This is exactly the statement in your question, if setting parentheses around "something" denotes the set of all such "something".

Answer 2

In this context, it seems to be the set of all mappings, which you can see because on the right of the equation you have $A \rightarrow (B \rightarrow C)$, and this wouldn't really make sense if $B \rightarrow C$ were a single mapping. I think $B \rightarrow C$ is usually written $C^B$. This 'equation' is called currying a function, where if we take a function $f(a, b) = \phi_{a,b}$ with two arguments, we can turn it into a function $g(a)$ which yields a function $g_a(b) = \phi_{a,b}$. It can be expressed with a universal property in the category of sets.

Answer 3

On page 15, around the middle you can find the following:

$2.22$. Definition. For any two sets $A$, $B$, $$(A\to B)=_{df}\{f\mid f\colon A\to B\}=\text{ the set of all functions from }A\text{ to }B.$$

So $(A\times B\to C)=_c(A\to(B\to C))$ simply means that the set of functions from $A\times B$ to $C$ has the same cardinality as the set of functions from $A$, to functions from $B$ to $C$.

The last set, $(A\to(B\to C))$ is a set of functions $f\colon A\to(B\to C)$ such that $f(a)$ is a function from $B$ to $C$.

(The question you asked if problem $\rm x2.7$.)

Answer 4

The notation $X \to Y$ usually denotes the type of functions which has domain $X$ and codomain $Y$. In context of set theory, that would be just the set of all functions from $X$ to $Y$, sometimes written also as $Y^X$ or ${}^XY$ to avoid order confusion (note the inversion in the first).

Hence, using the alternate notation, $A \times B \to C$ is $C^{A \times B}$, and $A \to (B \to C)$ is $(C^B)^A$.
The similarity with property of real numbers $(c^b)^a = c^{a\cdot b}$ is not a coincidence.

Relevant operations are commonly known as currying and uncurrying, should you have problems, you can check below.

Currying:

\begin{align}\mathrm{curry}(f) = x \mapsto y \mapsto f(x,y),\end{align}
or in different notation,
\begin{align}\mathrm{curry}(f)(x)(y) = f(x,y).\end{align}

Uncurrying:

\begin{align}\mathrm{uncurry}(f) = (x,y) \mapsto f(x)(y),\end{align} or written in the alternate way, \begin{align}\mathrm{uncurry}(f)(x,y) = f(x)(y).\end{align}

I hope this helps $\ddot\smile$