(I'm going to link to this Stackexchange post concerning Nullary Operations: [why is a nullary operation a special element, usually 0 or 1?](https://math.stackexchange.com/questions/273330/why-is-a-nullary-operation-a-special-element-usually-0-or-1))

In general an [operation](https://en.wikipedia.org/wiki/Operation_(mathematics)) is a function $f:S^n \to S$, where $n$ denotes the arity of the operation, and a *[nullary operation](https://en.wikipedia.org/wiki/Arity#Nullary)* is a function $f:S^0 \to S$.

It is clear that $S^n=S \times S \times \cdots \times S$ denotes the Cartesian product of a set with itself ($n$ times), however $S^0$ makes little sense to me.

One might define $S^0=\emptyset$. At first glance, that seems like a good choice. However then a nullary operation makes no sense, since it must be a function $f: A \to B$ where $A$ and $B$ are two sets representing the domain and codomain of the function. And note a function is defined in terms of *relation*, where a relation is a subset of the Cartesian product $A \times B$. So when we think of $A=S^0=\emptyset$ and $B=S$, so that we have a nullary operation $f:\emptyset \to S$, what is a a subset of $\emptyset \times S$? What even **is** $\emptyset \times S$? By definition of the Cartesian product,
\begin{equation}
\emptyset \times S = \{(a,b):a \in \emptyset\textrm{ and }b \in S\}.
\end{equation}
But there exists no $a \in \emptyset$! Therefore there exist no ordered pairs in our Cartesian product! Therefore $\emptyset \times S = \emptyset !$. Remember a function is a relation which is a subset of the Cartesian product. Therefore our relation is the empty set, and therefore also $f=\emptyset$.

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**Edit**: We got the definition sorted out in the comments. Thanks especially to Hayden and Christoff for their great help. In summary, we figured out that $S^0=\{()\}$, the set containing only one element, the "empty $n$-tuple". Then if taking $S=\{1,2,...,k\}$, then a nullary operation $f:S^0 \to S$ is a function, i.e. some subset of $S^0 \times S = \{()\} \times S$. Looking at $\{()\} \times S$, this equals $\{((),1),((),2),...,((),k)\}$. Then a *function* is a special subset of this, so that each element of the domain (and there is only one element, namely $()$) is paired with exactly one element of the codomain. Therefore a function would have the form $f=\{((),a)\}$ where $a$ is one of $1,2,...,k$. Thank you for your help.