If $S$ and $T$ are sets, which of the following would you say is $T^{S}$?
- $T^{S}$ is the set of all functions from $S$ to $T$.
- $T^{S}$ is the Cartesian product of $T$ taken $\begin{vmatrix} S \end{vmatrix}$ times. That is, $\underbrace{T \times T \times \cdots \times T \times T}_{\begin{vmatrix} S \end{vmatrix} \text{ times}}$?
As a simple example, suppose that ...
$T = \begin{Bmatrix} 0, 1 \end{Bmatrix}$.
If ...
$T^{T}$ is $\underbrace{T \times \cdots \times T}_{\begin{vmatrix} T \end{vmatrix} \text{ times}}$
... then ...
$T^{T} = \begin{Bmatrix} \begin{pmatrix} 0, 0 \end{pmatrix}, \begin{pmatrix} 0, 1 \end{pmatrix}, \begin{pmatrix} 1, 0 \end{pmatrix}, \begin{pmatrix} 1, 1 \end{pmatrix} \end{Bmatrix}$
However, suppose that $T^{T}$ is the set of all functions from set $T$ to set $T$.
Then $T^{T} = \{ f_{1}, f_{2}, f_{3}, f_{4} \} $ where ...
$$ \begin{align} f_{1} & = \{ (0, 0), (1, 0) \} \\ f_{2} & = \{ (0, 0), (1, 1) \} \\ f_{3} & = \{ (0, 1), (1, 0) \} \\ f_{4} & = \{ (0, 1), (1, 1) \} \\ \end{align} $$
An answer to this question would explain the basic notation used for cardinal arithmetic and explain all of the notation given in the formula(s) below:
$$\begin{align}A\rightarrow (B\rightarrow C)&\cong(B\rightarrow C)^A \\&\cong\left(C^B\right)^A \\&\cong C^{A\times B} \\&\cong(A\times B)\rightarrow C\end{align}$$
Is $\begin{pmatrix} B\rightarrow C \end{pmatrix}$ the set of all ordered pairs such that the left-most element in the pair comes from set $B$ and the right-most element in the pair comes from set $C$?
That is, $\begin{pmatrix} B\rightarrow C \end{pmatrix} = \begin{Bmatrix} (b, c) : b \in B \text{ and } c \in C\end{Bmatrix}$
For example, $\begin{pmatrix} B \rightarrow C \end{pmatrix}^{2} = \begin{Bmatrix} \begin{pmatrix} (b_{1}, c_{1}), (b_{2}, c_{2}) \end{pmatrix} : b_{1}, b_{2} \in B \text{ and } c_{1}, c_{2} \in C\end{Bmatrix}$
Alternatively, $\begin{pmatrix} B \rightarrow C \end{pmatrix}^{2} = \begin{Bmatrix} \begin{pmatrix} b_{1}, c_{1}, b_{2}, c_{2} \end{pmatrix} : b_{1}, b_{2} \in B \text{ and } c_{1}, c_{2} \in C\end{Bmatrix}$
In your explanation of the basics of the notation used for cardinal arithmetic you can assume that:
- we already know what a set is. For example { 1, 2, 3 }
- We know what a set of ordered pairs is. { (0, 1), (2, 1) }
- We know what a Cartesian product of two sets is.
- What already know what cardinality is. $4 = \vert \{ 1, 2, 3, 4 \} \vert$ and there is no bijection between the set of all whole numbers $\begin{pmatrix} \mathbb{N} \end{pmatrix}$ and the set of all decimal numbers $\begin{pmatrix} \mathbb{R} \end{pmatrix}$
As an answer to your first question: $T^{S}$, where $T$ and $S$ are sets, is referred to as all functions $f: S \rightarrow T$. The cardinality of $T^{S}$ is equal to $|T|^{|S|}$, if all sets in a cartesian product are the same then $T^n = \underbrace{T \times T \times \dots T}_\text{n times}$. Which is not quite the same as $T^{S}$ where $T$ and $S$ are sets.
Here is a link for reference and some more reading if you'd like: Sets, Functions and Relations