# Ternary operation

In mathematics, a **ternary operation** is an *n*-ary operation with *n* = 3. A ternary operation on a set *A* takes any given three elements of *A* and combines them to form a single element of *A*.

In computer science, a **ternary operator** is an operator that takes three arguments as input and returns one output.^{[1]}

## Examples[edit]

The function is an example of a ternary operation on the integers (or on any structure where and are both defined). Properties of this ternary operation have been used to define planar ternary rings in the foundations of projective geometry.

In the Euclidean plane with points *a*, *b*, *c* referred to an origin, the ternary operation has been used to define free vectors.^{[2]} Since (*abc*) = *d* implies *a* – *b* = *c* – *d*, these directed segments are equipollent and are associated with the same free vector. Any three points in the plane *a, b, c* thus determine a parallelogram with *d* at the fourth vertex.

In projective geometry, the process of finding a projective harmonic conjugate is a ternary operation on three points. In the diagram, points *A*, *B* and *P* determine point *V*, the harmonic conjugate of *P* with respect to *A* and *B*. Point *R* and the line through *P* can be selected arbitrarily, determining *C* and *D*. Drawing *AC* and *BD* produces the intersection *Q*, and *RQ* then yields *V*.

Suppose *A* and *B* are given sets and is the collection of binary relations between *A* and *B*. Composition of relations is always defined when *A* = *B*, but otherwise a ternary composition can be defined by where is the converse relation of *q*. Properties of this ternary relation have been used to set the axioms for a heap.^{[3]}

In Boolean algebra, defines the formula .

## Computer science[edit]

In computer science, a ternary operator is an operator that takes three arguments (or operands).^{[1]} The arguments and result can be of different types. Many programming languages that use C-like syntax^{[4]} feature a ternary operator, `?:`

, which defines a conditional expression. In some languages, this operator is referred to as the *conditional operator*.

In Python, the ternary conditional operator reads `x if C else y`

. Python also supports ternary operations called array slicing, e.g. `a[b:c]`

return an array where the first element is `a[b]`

and last element is `a[c-1]`

.^{[5]} OCaml expressions provide ternary operations against records, arrays, and strings: `a.[b]<-c`

would mean the string `a`

where index `b`

has value `c`

.^{[6]}

The multiply–accumulate operation is another ternary operator.

Another example of a ternary operator is *between*, as used in SQL.

The Icon programming language has a "to-by" ternary operator: the expression `1 to 10 by 2`

generates the odd integers from 1 through 9.

In Excel formulae, the form is =if(C, x, y).

## See also[edit]

- Median algebra or Majority function
- Ternary conditional operator for a list of ternary operators in computer programming languages
- Ternary Exclusive or
- Ternary equivalence relation

## References[edit]

- ^
^{a}^{b}MDN, nmve. "Conditional (ternary) Operator".*Mozilla Developer Network*. Retrieved 20 February 2017. **^**Jeremiah Certaine (1943) The ternary operation (abc) = a b^{−1}c of a group, Bulletin of the American Mathematical Society 49: 868–77 MR0009953**^**Christopher Hollings (2014)*Mathematics across the Iron Curtain: a history of the algebraic theory of semigroups*, page 264, History of Mathematics 41, American Mathematical Society ISBN 978-1-4704-1493-1**^**Hoffer, Alex. "Ternary Operator".*Cprogramming.com*. Retrieved 20 February 2017.**^**"6. Expressions — Python 3.9.1 documentation".*docs.python.org*. Retrieved 2021-01-19.**^**"The OCaml Manual: Chapter 11 The OCaml language: (7) Expressions".*ocaml.org*. Retrieved 2023-05-03.

## External links[edit]

- Media related to Ternary operations at Wikimedia Commons