It is clear to me that functions are relations, where a function $f(x) = y$ can be represented by a relation $xRy$.
What about functions with two arguments? For example $f(x,y)=z$. I know that we cannot treat this as $(x,y)Rz$, as we are not sure if $(x,y)$ actually is a relation or not. If there is such definition, how can I interpret some of the properties of relations such as symmetry, reflexive and transitivity?
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If the domain of the function is $X_1 \times X_2$ and the range is $Y$, then a function $f : X_1 \times X_2 \rightarrow Y$ is simply a relation $R$ that is a subset of $(X_1 \times X_2) \times Y$. Hence you get a relation between pairs $(x_1,x_2) \in X_1 \times X_2$ and values $y \in Y$.
More generally, functions of $n$ variables can be seen as subsets of $(X_1 \times X_2 \times \dots \times X_n) \times Y$.
Yes, it is a 3-place relation. The infix notation $(x,y)Rz$ is very awkward though, so typically we use something like $R(x,y,z)$