# What is the difference between functions and operations?

Wikipedia says that

an operation $\omega$ is a function of the form $\omega: V \to Y$, where $V \subset X_1 \times\cdots\times X_k$.

But as far as I know, every function's domain is a set, so every function can be seen as an operation where $V \subset X_1$. Thus, all functions are operations. Since all operations are also functions by the definition given, the terms "function" and "operation" are equivalent. What am I missing? Or are they truly equivalent?

• Any function can be seen as a unary operation over its domain. Jun 24, 2015 at 8:56

While your reasoning is correct that "every function is an operation" under that extremely general definition of "operation", I would say that a more common definition of an "operation" on a set $S$ would be a function $$\alpha: S^n\to S\quad\text{ for some }n\geq 0$$ or, to allow "partial" operations, $$\alpha: X\to S\quad\text{ where }X\subset S^n\text{ for some }n\geq 0$$ (and we would say $\alpha$ is an $n$-ary operation). Under this definition, there are clearly many functions that are not operations.