# Ternary operation

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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.[1]

## Examples

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

If F is a field, the function ${\displaystyle T(a,\ b,\ c)\ =\ ab+c}$ is an example of a ternary operator on F. 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 ${\displaystyle [a,b,c]\ =\ a-b+c}$ has been used to define free vectors.[2] Since (abc) = d implies ab = cd, 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 ${\displaystyle {\mathcal {B}}(A,B)}$ 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 ${\displaystyle [p,\ q,\ r]\ =\ pq^{T}r\ \ {\text{where}}\ \ q^{T}}$ 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, ${\displaystyle T(A,B,C)=AC+(1-A)B}$ defines the formula ${\displaystyle (A\lor B)\land (\lnot A\lor C)}$.

## Computer science

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.

## References

1. ^ a b MDN, nmve. "Conditional (ternary) Operator". Mozilla Developer Network. MDN. Retrieved 20 February 2017. CS1 maint: discouraged parameter (link)
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. Cprogramming.com. Retrieved 20 February 2017. CS1 maint: discouraged parameter (link)
5. ^ "6. Expressions — Python 3.9.1 documentation". docs.python.org. Retrieved 2021-01-19.
6. ^ "7.7 Expressions". caml.inria.fr. Retrieved 2021-01-19.