Ternary operation

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The function $T(a,\; b,\; c)\; =\; ab\; +\; c$ is an example of a ternary operation on the integers (or on any structure where $+$ and $\backslash times$ 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 $[a,\; b,\; c]\; =\; a\; -\; b\; +\; c$ has been used to define free vectors.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⁻¹c of a group], Bulletin of the American Mathematical Society 49: 868–77 {{MR|id=0009953}} 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 $\backslash mathcal\{B\}(A,\; B)$ 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 $[p,\; q,\; r]\; =\; p\; q^T\; r$ where $q^T$ is the converse relation of q. Properties of this ternary relation have been used to set the axioms for a heap.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}}

In Boolean algebra, $T(A,B,C)\; =\; AC+(1-A)B$ defines the formula $(A\; \backslash lor\; B)\; \backslash land\; (\backslash lnot\; A\; \backslash lor\; C)$.

In computer science, a ternary operator is an operator that takes three arguments (or operands). The arguments and result can be of different types. Many programming languages that use C-like syntax{{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}} 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].{{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}} OCaml expressions provide ternary operations against records, arrays, and strings: a.[b]<-c would mean the string a where index b has value c.{{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}}

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).

• 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

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• {{Commons category-inline|Ternary operations}}