Ternary Operation

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 [math]\displaystyle{ T(a, b, c) = ab + c }[/math] is an example of a ternary operation on the integers (or on any structure where [math]\displaystyle{ + }[/math] and [math]\displaystyle{ \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]\displaystyle{ [a, b, c] = a - b + c }[/math] 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 [math]\displaystyle{ \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]\displaystyle{ [p, q, r] = p q^T r }[/math] where [math]\displaystyle{ q^T }[/math] 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, [math]\displaystyle{ T(A,B,C) = AC+(1-A)B }[/math] defines the formula [math]\displaystyle{ (A \lor B) \land (\lnot A \lor C) }[/math].

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

• Median algebra
• Ternary conditional operator for a list of ternary operators in computer programming languages

References

External links

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