Definition:Operation

Informal Definition
An operation (or operator) is an object, identified by a symbol, which can be interpreted as a process which, from a number of objects, creates a new object.

Formal Definition
An operation is a mapping $\circ$ from a cartesian product of $n$ sets $S_1 \times S_2 \times \ldots \times S_n$ to a universal set $\mathbb U$:


 * $\circ: S_1 \times S_2 \times \ldots \times S_n \to \mathbb U: \forall \left({s_1, s_2, \ldots, s_n}\right) \in S_1 \times S_2 \times \ldots \times S_n: \circ \left({s_1, s_2, \ldots, s_n}\right) = t \in \mathbb U$

An operation needs to be defined for all tuples in $S_1 \times S_2 \times \ldots \times S_n$.

Arity
The number of operands an operator takes is called its arity or its valency.

Operation on a Set
An $n$-ary operation on a set $S$ is an operation where the domain is the cartesian space $S^n$ and the codomain is $S$:


 * $\circ: S^n \to S: \forall \left({s_1, s_2, \ldots, s_n}\right) \in S^n: \circ \left({s_1, s_2, \ldots, s_n}\right) = t \in S$

An $n$-ary operation on $S$ needs to be defined for all tuples in $S^n$.

Operand
An operand is one of the object on which an operator generates its new object.

Binary Operation
A binary operation is the special case of an operation where the operation has exactly two operands.

A binary operation is a mapping $\circ$ from the Cartesian product of two sets $S \times T$ to a universe $\mathbb U$:


 * $\circ: S \times T \to \mathbb U: \circ \left ({s, t}\right) = y \in \mathbb U$

If $S = T$, then $\circ$ can be referred to as a binary operation on $S$.

Note that a binary operation is a special case of a general operator, i.e. one that has two operands.

If $\circ$ is a binary operation on $S$, then for any $T \subseteq S$, $\circ \left ({x, y}\right)$ is defined for every $x, y \in T$. So $\circ$ is a binary operation on every $T \subseteq S$.

Infix Notation
A far more common alternative to the notation $\circ \left ({x, y}\right) = z$, which works for a binary operation, is to put the symbol for the operation between the two operands: $z = x \circ y$.

This is called infix notation.

Product
For a given operation $\circ$, let $z = x \circ y$.

Then $z$ is called the product of $x$ and $y$.

This is an extension of the normal definition of product that is encountered in conventional arithmetic.

Unary Operation
A unary operation is the special case of an operation where the operation has exactly one operand.

Thus, a unary operation on a set $S$ is a mapping whose domain and codomain are both $S$.

Comment
It can be seen that, in the same way that a mapping can be seen as a way of "transforming" one element into to another, an operation does the same thing, just with a larger number of operands.

In fact, as we have just defined it, we see that an operation is a generalisation of the concept of the mapping, or (if you like) a mapping is just an operation with only one operand.

There is another way to view an operation. Instead of viewing it as the act of combining two things in a certain way to get a third, we can look upon it as doing something to the first thing with the second to turn it into the third.

Thus, $\circ \left ({a, b}\right)$ can be interpreted as $\circ_b \left ({a}\right)$, where $\circ_b$ is defined as the mapping which performs "$\circ_b$" on a single operand.

For example, take the statement "$1 + 2 = 3$", where the symbol $+$ represents the familiar binary operation of addition of numbers. Thus, we can either view $+$ as being the operation that takes $1$ and $2$ and maps them onto $3$, or we can say that we take $1$, and then we do something to it: we "add $2$", and this turns the $1$ into $3$.

In the case of addition, in a certain sense the first interpretation comes to mind more easily than the second, but if we take the statement "$3 - 2 = 1$", it's more natural to think of this as "doing something" to $3$, that is, to take $2$ off it, to change it into something smaller, that is, $1$.

Both interpretations are equally valid, but depending on the circumstances, one may be more appropriate than the other.

Examples
An example of an operator, from conventional arithmetic, is "$+$", as in, for example, "$2 + 3 = 5$". The operands (in this particular instance) are $2$ and $3$.