What does it mean when a set is the exponent? Can you help me understand what the exponent means here?
$$ (A\cap B) \cup D^C $$
A, B, C and D are all sets of a universal set U.
Thank you.
 A: Small lower-case $c$ as an exponent when a Universe of Discourse is understood generally represents complement relative to $U$. That seems to be almost certainly the meaning in what you write: $(A\cap B)\cup D^c$ would be the collection of all things that are in either in both $A$ and $B$, or else that are not in $D$, as
$$D^c = \{x\in U \mid x\notin D\}.$$
If that's the case (a lower-case $c$), then $D^c$ is pronounced "complement of $D$" or "$D$-complement."
Note the difference between
$$D^c\quad\text{and}\quad D^C.$$
If $C$ is a set, then $D^C$ is, as others have noted, the set of all functions with domain $C$ and image contained in $D$,
$$D^C = \{f\colon C\to D\mid f\text{ is a function}\}.$$
A: Generally, when we write $D^C$ for two sets $D,C$, what we mean is the set of functions $f:C \rightarrow D$.  Consider for example $C$ being a two-element set, $\{ 1,2 \}$.  Then the set of functions from $C$ to $D$ is just the set of ordered pairs $(d_1,d_2)$ of elements of $D$ -- and hence has cardinality $|D|^2$.  In general, the cardinality of $D^C$ is $|D|^{|C|}$ for finite sets, and we use that to generalize cardinality exponentiation for infinite sets.
A: In the general set theoretic settings, $X^Y=\{f\colon Y\to X\mid f\ \text{is a function}\}$. That is, this is the set of all functions with domain $Y$ and range which is a subset of $X$.
In the introductory context, in which there is usually a universal set being involved (although these can be involved in an explicit way much further into mathematics, however I doubt you would have asked this question in that situation), it is common to denote with a lowercase $c$ the complement with respect to the universal set, that is $D^c=U\setminus D=\{x\in U\mid x\notin D\}$, where $U$ is the universal set.
