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I am studying, as a self-taught, the book Analysis I, by Terence Tao.

I note that the author sometimes uses the word 'operation', but never defines it.

Incidentally, the word 'operation' does not even appear in Index.

On page 16, some operations are mentioned: addition, multiplication, exponentiation, and especially the increment operation.

On page 49, preceding the function definition, you can read:

"Informally, a function f: X ---> Y from one set X to another set Y is an operation which assigns to each element (or" input ") x in X, a single element (or" output ") f(x) in Y."

At the bottom of page 329, you can read:

"(Substitution axiom). Given any two objects x and y of the same type, if x = y, then f(x) = f(y) for all functions or operations f."

I do not know if the author distinguishes between function and operation. I need help.

Paulo Argolo
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  • Usually a binary operation on a set $X$ is a function $X \times X \to X$. –  Aug 28 '17 at 19:55
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    No, there is no distinction. He uses "operation" informally, with "function" being the officially defined term. The intention of using the term "operation" is to convey the idea of a process. But of course a function needs not be given by any describable set of rules. – Andrés E. Caicedo Aug 28 '17 at 19:55
  • Yes; "operation" is an "informal" mathematical term for function (usually: unary or binary). – Mauro ALLEGRANZA Sep 12 '17 at 07:31

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Formally, an operation on a set $X$ is a function $f:X^n\to X$. For example, addition is a function $+:\mathbb{R}^2 \to \mathbb{R}$.

In the first example you give, where Tao explicitly says that he's being informal, "operation" is just used to connote some sort of function/relation between two sets without being too specific. He's trying to provide some intuition behind the more formal definitions that follow.

cderwin
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