Definition:Operation

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Definition

An operation is an object, identified by a symbol, which can be interpreted as a process which, from a number of objects, creates a new object.


n-Ary Operation

Let $S_1, S_2, \dots, S_n$ be sets.

Let $\circ: S_1 \times S_2 \times \ldots \times S_n \to \mathbb U$ be a mapping from the cartesian product $S_1 \times S_2 \times \ldots \times S_n$ to a universal set $\mathbb U$:

That is, suppose that:

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


Then $\circ$ is an $n$-ary operation.


Arity

The arity of an operator is the number of operands it uses.


The arity of an operator may be, in general, any number.

It may even be infinite.


Operation on a Set

An $n$-ary operation on a set $S$ is an $n$-ary operation where:

the domain is the cartesian space $S^n$
the codomain is $S$:
$\odot: S^n \to S: \forall \left({s_1, s_2, \ldots, s_n}\right) \in S^n: \odot \left({s_1, s_2, \ldots, s_n}\right) \in S$


That is:

an $n$-ary operation on $S$ needs to be defined for all tuples in $S^n$
the image of $\odot$ is itself in $S$.


Operand

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


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 $S$.


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 $\mathbb U$:

$\circ: S \times T \to \mathbb U: \circ \tuple {s, t} = y \in \mathbb U$

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


Also known as

An operation is also known as an operator.


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 operation, from conventional arithmetic, is "$+$", as in, for example, $2 + 3 = 5$.

The operands (in this particular instance) are $2$ and $3$.


Sources