{{Short description|Mathematical operation that combines three elements to produce another element}} 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.<ref name = "MDM nmve">{{cite web |last1=MDN |first1=nmve |title=Conditional (ternary) Operator |website=Mozilla Developer Network |url=https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Operators/Conditional_Operator |accessdate=20 February 2017}}</ref>

==Examples== thumb|right|Given ''A'', ''B'' and point ''P'', geometric construction yields ''V'', the projective harmonic conjugate of ''P'' with respect to ''A'' and ''B''.

The function <math>T(a, b, c) = ab + c</math> is an example of a ternary operation on the integers (or on any structure where <math>+</math> and <math>\times</math> 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 <math>[a, b, c] = a - b + c</math> has been used to define free vectors.<ref>Jeremiah Certaine (1943) [https://www.ams.org/journals/bull/1943-49-12/S0002-9904-1943-08042-1/S0002-9904-1943-08042-1.pdf The ternary operation (abc) = a b<sup>−1</sup>c of a group], Bulletin of the American Mathematical Society 49: 868–77 {{MR|id=0009953}}</ref> Since (''abc'') = ''d'' implies ''b'' – ''a'' = ''c'' – ''d'', the directed line segments ''b'' – ''a'' and ''c'' – ''d'' 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 <math>\mathcal{B}(A, B)</math> 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 <math>[p, q, r] = p q^T r</math> where <math>q^T</math> is the converse relation of ''q''. Properties of this ternary relation have been used to set the axioms for a heap.<ref>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}}</ref>

In Boolean algebra, <math>T(A,B,C) = AC+(1-A)B</math> defines the formula <math>(A \lor B) \land (\lnot A \lor C)</math>.

==Computer science==

In computer science, an operator is a ternary operator if it takes three arguments (or operands).<ref name="MDM nmve"/>

Many programming languages that use C-like syntax<ref>{{cite web |last1=Hoffer |first1=Alex |title=Ternary Operator |website=Cprogramming.com |url=https://www.cprogramming.com/reference/operators/ternary-operator.html |accessdate=20 February 2017}}</ref> feature the ternary conditional operator, <code>?:</code>, which defines a conditional expression that yields a value. This is sometimes referred to simply as the ternary operator, despite that several unrelated ternary operators exist.

In the expression <code>x = a ? b : c</code> the variable ''x'' will be assigned the value ''b'' if ''a'' is true. Otherwise it will be assigned the value ''c''.

Some languages use a different syntax. In Python, the same expression would take the form <code>x = b if a else c</code>. In Excel formulae, the form is <code>=IF(a, b, c)</code>.

Many languages do not have a ternary conditional operator, though some have an alternative. For instance, although Ruby does have the ternary conditional operator, its <code>if/elsif/else</code> flow control structure yields a value, so it can serve the same purpose. In SQL, the <code>CASE</code> expression evaluates many conditionals to yield a value. These examples are not strictly ternary because they may have more than three components.

Ternary operators other than the ternary conditional operator exist.

In Python the expression <code>a[b:c]</code> will slice a portion of an array. The result is a new array containing all the elements of ''a'' from ''b'' to ''c-1''.<ref>{{cite web |title=6. Expressions — Python 3.9.1 documentation |url=https://docs.python.org/3/reference/expressions.html |access-date=2021-01-19 |website=docs.python.org}}</ref>

In OCaml the expression <code>a.(b) <- c</code> updates element ''b'' of array ''a'' to value ''c''.<ref>{{cite web |title=The OCaml Manual: Chapter 11 The OCaml language: (7) Expressions |website=ocaml.org |url=https://v2.ocaml.org/manual/expr.html |access-date=2023-05-03}}</ref>

In some assembly languages the MAD operation is in ternary form. The statement <code>MAD a, b, c</code> multiplies ''b'' and ''c'', adds the result to ''a'', and stores the final result in ''a'', all in a single CPU cycle. In some assembly languages the order of the operands may differ. In some, the operation isn't ternary because it requires a fourth operand to indicate the location where the result will be stored.

The SQL expression ''BETWEEN'' is ternary, as in <code>age BETWEEN 90 AND 100</code>.

The Icon expression ''to'' becomes ternary when used with ''by'', as in <code>1 to 10 by 2</code>, which generates the odd integers from 1 through 9.

==See also== * Unary operation * Unary function * Binary operation * Iterated binary operation * Binary function * 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== {{Reflist}}

==External links== *{{Commons category-inline|Ternary operations}}

Category:Ternary operations