Set theory pertaining to category theory and functional programming

Question

I'm reading an unfinished Introduction to Category Theory/Products and Coproducts of Sets and have come across the following:

A power set of a set is the set of all its subsets. A script 'P' is used for the power set. Note that the empty set and the set itself are members of the power set. \begin{equation}\mathcal{P}\{1,2,3\} = \{\varnothing,\{1\},\{2\},\{3\},\{1,2\},\{1,3\},\{2,3\},\{1,2,3\}\}\end{equation} Set operations are functions from the product of a power set to a power set. \begin{equation}\text{Union}:\mathcal{P}(X) \times \mathcal{P} \to \mathcal{P}(X), \text{Union}(A,B) = A \cup B\end{equation}

I don't understand the last part.

To me a product of a power set should equate to a set of tuples where the first of the ordered pair is a subset from first power set and the second is a subset from the second power set. Whereas a power set is just a set of subsets.

How can this mismatch be explained or is it a mistake?

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Taking on board the advice below:

If the set $X$ is a singleton value $1$, then: \begin{equation}X = \{1\}\end{equation} The power set of $X$ is then: \begin{equation}\mathcal{P}(X) = \{\varnothing,\{1\}\}\end{equation} The product of $\mathcal{P}(X)$ is then: \begin{equation}\mathcal{P}(X)\times\mathcal{P}(X) = \{(\varnothing,\varnothing),(\varnothing,\{1\}),(\{1\},\varnothing),(\{1\},\{1\})\}\end{equation} Applying $\text{Union}$ to the $\mathcal{P}(X)\times\mathcal{P}(X)$ is: \begin{equation}\mathcal{P}(X)\times\mathcal{P}(X) = \{(\varnothing,\varnothing)\mapsto\varnothing,(\varnothing,\{1\})\mapsto\{1\},(\{1\},\varnothing)\mapsto\{1\},(\{1\},\{1\})\mapsto\{1\}\}\end{equation} As the result is a set, duplicates can be removed: \begin{equation}\{\varnothing,\{1\},\{1\},\{1\}\} = \{\varnothing,\{1\}\}\end{equation} Thus: \begin{equation}\mathcal{P}(X)\times\mathcal{P}(X) \to\mathcal{P}(X)\end{equation} And so the set operation of $\text{Union}$, which is one of a family of set operations (a set of functions) is a function from the product of a power set to a power set

Answer 1

The notation $f:E\times F \to G$ means that $f$ is a function that needs two arguments, one from $E$, one from $F$, and the image is in $G$.

This is how the function $\text{Union}$ is defined: the two arguments $A$, $B$ are in $\mathcal{P}(X)$ and the image $\text{Union}(A, B) = A\cup B$ is in $\mathcal{P}(X)$.

Answer 2

As you have said, $X\times Y = \{(x,y) \mid x\in X, y\in Y\}$.

Thus, a function $f:X\times Y\rightarrow Z$ would get two arguments: one from $X$ and the other from $Y$, and output a value from $Z$. Formally, this is written as $f((x,y))=z$, and to reduce the number of brackets, its usually written as just $f(x,y)=z$.

In your case, $f$ is a function that computes the union: It takes two elements from $A,B\in P(X)$ and outputs $A\cup B\in P(X)$.

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For clarity, the formal definition of $P(X)$ is given by:

\begin{equation}P(X):=\{A\mid A\subseteq X\}\end{equation}

Is the set of all subsets of $X$.

Answer 3

The product of a power set to a power set is, indeed, "a set of tuples where the first of the ordered pair is a subset from first power set and the second is a subset from the second power set", to use your own words.

What you reported, though, is the definition of the Union operation. You can think of it as a function that associates an element of the set "tuples..." to an element of the set P(x). In particular, the Union operator, when applied to a pair of subsets, returns a single subset that is the union of the two subsets in the tuple.