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 $$3$$.