# Exponential Notation

Let’s enjoy some familiar concepts in sets and be happy with the exponential notation of function space.

Normally, a function from set X to set Y is represented as $f:X \rightarrow Y$. Sometimes, people say that it is the exponential object in the category of Set, but what does it mean? Is it similar to the exponentiation of numbers, e.g. $2^3$?

Indeed, yes.

We can view the disjoint union and intersection as addition and multiplication respectively. For example, we have

$\displaystyle{ (\mathbf{2} \rightarrow A) \cong A \times A}$

where $\mathbf{2}$ denotes the two-pointed set. In face, the isomorphism is given by $f \mapsto (f(0), f(1))$ and $(x, y) \mapsto g$ where $g(0) = x$ and $g(1) = y$. In general, it means that

$\displaystyle{(B \rightarrow A) \cong \underbrace{A \times A \times \dots \times A}_{| B |}}$.

Using exponential notation, it reads that $A^{B} \cong \prod_{i = 1}^{|B|} A$. It satisfies familiar equalities we learned, e.g.

1. $A^B \times A^C \cong A^{B + C}$.
2. $(A^{B})^{C} \cong A^{B \times C}$.
3. $A^{0} \cong 1$ where $0$ is the empty set and $1$ is the one-pointed set.
4. $0^{A} \cong 0$ for any non-empty set $A$.

With this notation, we can drop the subset notation $\subseteq$ since $S \subseteq X$ is equivalent to $S \in \mathbf{2}^{X}$ where $S$ becomes the characteristic function on $S$. That is,

$\chi_{S}(x) = 1$ if $x \in S$ or $\chi_{S}(x) = 0$ otherwise.

Moreover, the belonging relation $\in$ can be viewed as a function from one-pointed set to $X$, e.g. $(* \rightarrow A) \equiv A^{*}$. Note that $x \in X$ is a proposition in set theory but a function $x : * \rightarrow A$ is a proposition in the meta-language.

Looking at the above discussion more carefully, we may suspect whether we can talk about set theory purely in terms of objects, morphisms, hom-set and so on. Well, indeed! Basically it is called topos theory and maybe I can say something about topos theory after I fully understand it …

## 4 thoughts on “Exponential Notation”

1. Interesting. I’d be interested learning something about topos.

And when will we learn that 2+2 = 4? 🙂

2. And when will we learn that 2+2 = 4?

Do you mean Peano axioms? or, the coproduct of 2 and 2 is 4 in $\mathbf{Finord}$?

Although I wrote that I would talk about topos, but it’s still far from the goal … Any idea to introduce the exponential object, sub-object and sub-object classifier?

3. XOO

難怪這個 2+2=4 好眼熟。XD

Okay, let me explain it to myself and write down what I have been explained!