8
$\begingroup$

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?

$\endgroup$
1
  • $\begingroup$ Any function can be seen as a unary operation over its domain. $\endgroup$ – Emilio Novati Jun 24 '15 at 8:56
3
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.