Template:Short description 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">{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>
ExamplesEdit
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) The ternary operation (abc) = a b−1c of a group, Bulletin of the American Mathematical Society 49: 868–77 Template:MR</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 Template:ISBN</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 scienceEdit
In computer science, a ternary operator is an operator that takes three arguments (or operands).<ref name="MDM nmve"/> The arguments and result can be of different types. Many programming languages that use C-like syntax<ref>{{#invoke:citation/CS1|citation
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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].<ref>{{#invoke:citation/CS1|citation
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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 alsoEdit
- 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