Ternary operation

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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.[1]

Examples

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)=ab+c 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 [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) 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]=pqTr where qT 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)=AC+(1−A)B defines the formula (A∨B)∧(¬A∨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.

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

See also

References

  1. ↑ 1.0 1.1 MDN, nmve. "Conditional (ternary) Operator". https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Operators/Conditional_Operator. 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". https://www.cprogramming.com/reference/operators/ternary-operator.html. Retrieved 20 February 2017. 
  5. ↑ "6. Expressions — Python 3.9.1 documentation". https://docs.python.org/3/reference/expressions.html. 
  6. ↑ "The OCaml Manual: Chapter 11 The OCaml language: (7) Expressions". https://v2.ocaml.org/manual/expr.html. 



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