Creating L-structures on domains I'm given the language $L=(R,g)$ and in this, $R$ is a relation of arity 3and $g$ is a function of arity 2. If I have the domain {1, 2, 3, 4}, how many different L-structures can I make with this domain? I have a side question: If the function is binary, does it send two inputs to one outputs or can it send it to two outputs? and for $g$, is $g(1,2)$ the same as $g(2,1)$? Thanks for your help

Let $D = \\{1,2,3,4\\}$ be your domain.

A relation symbol of arity $3$ is interpreted as a subset of $D^3$. What is the size of $D^3$? How many subsets does a set of size $k$ have?

A function symbol of arity $2$ is interpreted as a function $D^2 \to D$. What is the size of $D^2$? What is the size of $D$? How many functions are there from a set of size $m$ to a set of size $n$?

Answering the questions above should tell you how many choices you have for interpreting $R$ and interpreting $g$ on the domain $D$. Now an $L$-structure with domain $D$ consists of an interpretation of $R$ and an interpretation of $g$. How many choices do you have total?