Ternary operation

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.[1]

## Examples[edit]

Given A, B and point P, geometric construction yields V, the projective harmonic conjugate of P with respect to A and B.

The function T ( a , b , c ) = a b + c {\displaystyle T(a,b,c)=ab+c} is an example of a ternary operation on the integers (or on any structure where  + {\displaystyle +} and  × {\displaystyle \times } 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  [ a , b , c ] = a − b + c {\displaystyle [a,b,c]=a-b+c} 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  B ( A , B ) {\displaystyle {\mathcal {B}}(A,B)} 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  [ p , q , r ] = p q T r {\displaystyle [p,q,r]=pq^{T}r} where  q T {\displaystyle q^{T}} 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,  T ( A , B , C ) = A C + ( 1 − A ) B {\displaystyle T(A,B,C)=AC+(1-A)B} defines the formula  ( A ∨ B ) ∧ ( ¬ A ∨ C ) {\displaystyle (A\lor B)\land (\lnot A\lor C)} .

## 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]

1. ^ a b MDN, nmve. "Conditional (ternary) Operator". Mozilla Developer Network. Retrieved 20 February 2017.
2. ^ Jeremiah Certaine (1943) The ternary operation (abc) = a b−1c of a group, Bulletin of the American Mathematical Society 49: 868–77 MR0009953
3. ^ 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
4. ^ Hoffer, Alex. "Ternary Operator". Cprogramming.com. Retrieved 20 February 2017.
5. ^ "6. Expressions — Python 3.9.1 documentation". docs.python.org. Retrieved 2021-01-19.
6. ^ "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