# Definition:Operation

## Contents

## 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:

- $\circ: S_1 \times S_2 \times \ldots \times S_n \to \mathbb U: \forall \tuple {s_1, s_2, \ldots, s_n} \in S_1 \times S_2 \times \ldots \times S_n: \map \circ {s_1, s_2, \ldots, s_n} \in \mathbb U$

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 $\mathbb U$:

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

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

- 1996: H. Jerome Keisler and Joel Robbin:
*Mathematical Logic and Computability*... (previous) ... (next): Appendix $\text{A}.8$: Cartesian Product - 1999: András Hajnal and Peter Hamburger:
*Set Theory*... (previous) ... (next): $1$. Notation, Conventions: $11$