# Definition:Operation/Binary Operation

## Definition

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** $S$.

### Infix Notation

Let $\circ: S \times T \to U$ be a binary operation.

When $\map \circ {x, y} = z$, it is common to put the symbol for the operation between the two operands:

- $z = x \circ y$

This convention is called **infix notation**.

### Prefix Notation

Let $\circ: S \times T \to U$ be a binary operation.

The convention that places the symbol for the operation before the two operands:

- $z = \map \circ {x, y}$

is called **prefix notation**.

### Postfix Notation

Let $\circ: S \times T \to U$ be a binary operation.

The convention that places the symbol for the operation after the two operands:

- $z = \left ({x, y}\right) \circ$

is called **postfix notation**.

### Product

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.

## Also known as

Some authors use the term **(binary) composition** or **law of composition** for **(binary) operation**.

Some sources refer to a **binary operation** just as an **operation**.

Most authors use $\circ$ for **composition of relations** (which, if you think about it, is itself an **operation**) as well as for a general **operation**.

To avoid confusion, some authors use $\bullet$ for **composition of relations** to avoid ambiguity.

1965: Seth Warner: *Modern Algebra* uses $\bigtriangleup$ and $\bigtriangledown$ for the general **binary operation**, which has the advantage that they are unlikely to be confused with anything else in this context.

1975: T.S. Blyth: *Set Theory and Abstract Algebra* uses $\intercal$, and calls it **truc**, French for **trick** or **technique**:

*The symbol $\intercal$ is called***truc**("trook") and is French for "thingummyjig"! The idea it conveys is that what we call our law of composition does not matter, for what we are really interested in are sets of objects and mappings between them.

The $\LaTeX$ code for \(\intercal\) is `\intercal`

.

## Also defined as

Some authors specify that a **binary operation** $\circ$ is defined such that the codomain of $\circ$ is the same underlying set as that which forms the domain.

That is:

- $\circ: S \times S \to S$

and thus gloss over the fact that a **binary operation** defined in such a way is **closed**.

Such a treatment can obscure the detail of the development of the theory of algebraic substructures; **closedness** is a fundamental concept in this context.

## Also see

- Binary Operation on Subset is Binary Operation
- Definition:Iterated Binary Operation
- Definition:Unary Operation

## Sources

- 1951: Nathan Jacobson:
*Lectures in Abstract Algebra: Volume $\text { I }$: Basic Concepts*... (previous) ... (next): Introduction $\S 2$: Product sets, mappings - 1964: Iain T. Adamson:
*Introduction to Field Theory*... (previous) ... (next): Chapter $\text {I}$: Elementary Definitions: $\S 1$. Rings and Fields - 1964: W.E. Deskins:
*Abstract Algebra*... (previous) ... (next): $\S 1.4$: Definition $1.10$ - 1964: W.E. Deskins:
*Abstract Algebra*... (previous) ... (next): Exercise $1.4: \ 7$ - 1964: Walter Ledermann:
*Introduction to the Theory of Finite Groups*(5th ed.) ... (previous) ... (next): Chapter $\text {I}$: The Group Concept: $\S 1$: Introduction - 1964: William K. Smith:
*Limits and Continuity*... (previous) ... (next): $\S 2.1$: Sets - 1965: J.A. Green:
*Sets and Groups*... (previous) ... (next): $\S 4.1$. Binary operations on a set - 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 2$: Compositions - 1966: Richard A. Dean:
*Elements of Abstract Algebra*... (previous) ... (next): $\S 0.5$ - 1968: Ian D. Macdonald:
*The Theory of Groups*... (previous) ... (next): Appendix: Elementary set and number theory - 1969: C.R.J. Clapham:
*Introduction to Abstract Algebra*... (previous) ... (next): Chapter $1$: Integral Domains: $\S 2$. Operations - 1970: B. Hartley and T.O. Hawkes:
*Rings, Modules and Linear Algebra*... (next): Chapter $1$: Rings - Definitions and Examples: $1$: The definition of a ring - 1972: A.G. Howson:
*A Handbook of Terms used in Algebra and Analysis*... (previous) ... (next): $\S 2$: Sets and functions: Operations - 1973: C.R.J. Clapham:
*Introduction to Mathematical Analysis*... (previous) ... (next): Chapter $1$: Axioms for the Real Numbers: $2$. Fields - 1974: Thomas W. Hungerford:
*Algebra*... (next): $\text{I}$: Groups: $\S 1$ Semigroups, Monoids and Groups - 1975: T.S. Blyth:
*Set Theory and Abstract Algebra*... (previous) ... (next): $\S 11$ - 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): $\S 27$. Binary operations - 1982: P.M. Cohn:
*Algebra Volume 1*(2nd ed.) ... (previous) ... (next): $\S 3.1$: Monoids - 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $2$ - 1996: H. Jerome Keisler and Joel Robbin:
*Mathematical Logic and Computability*... (previous) ... (next): Appendix $\text{A}.8$: Cartesian Product - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $2$ - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**algebraic operation** - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**binary operation** - 2000: James R. Munkres:
*Topology*(2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 4$: The Integers and the Real Numbers - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**algebraic operation** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**binary operation** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**operation**