Why is $\text{Funct}(X, Y) \subseteq \mathcal P (X \times Y)$? I searched for "set of all functions power set" but didn't find anything that looked relevant.
Background:
I am struggling to understand this paragraph from page 21 of Analysis I by Amann and Escher.



The Remark cited is shown below.



Questions and comments:
I don't understand why the set of all functions from $X$ to $Y$ (i.e. $\text{Funct}(X, Y)$) is a subset of the set of all subsets of $X \times Y$ (i.e. $\mathcal P(X \times Y)$). It says in the Remark that a function from $X$ to $Y$ is a an ordered triple $(X, G, Y)$ meeting certain conditions. Doesn't that mean the set of all functions from $X$ to $Y$ should be the set of all those ordered triples?
I'm also struggling to understand the part explaining that $X^n$ is the set of all functions from $\{ 1, 2, \dots, n \}$ to $X$. To me, the $n$-fold Cartesian product of $X$ is all the ordered $n$-tuples you get by selecting any $x \in X$ for each of the $n$ positions in the $n$-tuple. A function from $\{ 1, 2, \dots, n \}$ to $X$ would have as its image just a single $n$-tuple $(x_1, \dots, x_n)$, right? I guess if you identify each $n$-tuple with a function, then the set of all $n$-tuples is the set of all functions?
Thanks for any help.
 A: The book’s treatment is sloppy. By their definition in Remark $\bf{3.1}$ a function from $X$ to $Y$ is not in fact a subset of $X\times Y$, despite what they say in their definition of $\operatorname{Funct}(X,Y)$. In that definition they’re actually talking about the set $G$ of ordered pairs, not the ordered triple $\langle X,G,Y\rangle$. The notation $\operatorname{Funct}(X,Y)$ identifies the domain and codomain of the functions in question, so there’s no loss of information in taking it to be the set of $G$s, but what they said is simply false.
$X^n$ can be defined as $\operatorname{Funct}(\{1,\ldots,n\},X)$, or it can be defined as the set of $n$-tuples of elements of $X$. These are formally distinct objects, but there is a natural bijection between them, so we normally just think of them as the same thing and use whichever representation is more convenient at the time.
A: My own view (and one that's pretty important in much of computer science) is that a function, like Gaul, is divided in three parts: $f = (D, C, R)$, where $R$ is a subset of $D \times C$ with certain properties to make it "function like". $R$ is what the author of your text would call the "graph" of the function. $D$ and $C$ are called (by some) the domain and codomain.
So reading the definition carefully, as you did, shows that the set of all functions isn't actually a subset of $P(X \times Y)$, but instead,
$$Funct(X, Y) \subset \{X\} \times \{Y\} \times P(X \times Y).$$
On the other hand, there's a function that I'll call $\pi_3$ (for "projection on the third component") that sends $f = (D, C, R)$ to $R$, which maps $Funct(X, Y)$ to $P(X \times Y)$, and the author is hiding this in the description (alas).
In short: your understanding is fine; the author is being glib.
A: You can identify a function $$f:X\rightarrow Y$$
with the set
$$\operatorname{graph}(f):=\{(x,f(x)):x\in X\}$$
which is a subset of $$X\times Y$$
Therefore the set $\operatorname{Funct}(X,Y)$ is a set of sets:
$$\operatorname{Funct}(X,Y)=\{f:X\rightarrow Y\}=\Big\{\{(x,f(x)):x\in X\}:f\text{ is a function}\Big\}\subseteq\mathcal{P}(X\times Y)$$
