Domain formalism

Recall that we can write the power set of $X$ as $2^X$, the set of all functions from $X$ to $\{0,1\}$, with each function $F \in 2^X$ defined as

\[ F = \{ X \times \{0,1\}: \text{ every $x \in X$ maps uniquely to }\{0,1\} \}\]

The ordered pairs defined by $X \times \{0,1\}$ actually amount to graphs or representations of $F$. Note that a presentation is a result, so that pairs here are representations of $F$ rather than presentations of $F$; on the other hand, the pairs are presentations of some other process, namely $2^X$ as a statement in set theory. Leaving these aside, the notation $2^X$ defines a set of functions with $X$ as domain or context and $\{0,1\}$ as target or result. Similarly, we use $\uparrow^\downarrow$ to suggest that the domain is defined or $\downarrow$ or while the result is undefined or $\uparrow$.