Ternary operation

Last updated December 03, 2024

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.

Examples

The function $T(a,b,c)=ab+c$ is an example of a ternary operation on the integers (or on any structure where $+$ and $\times$ 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.^[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 ${\mathcal {B}}(A,B)$ 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 $[p,q,r]=pq^{T}r$ where $q^{T}$ is the converse relation of q. Properties of this ternary relation have been used to set the axioms for a heap.^[3]

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

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.

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

Related Research Articles

In mathematics, the associative property is a property of some binary operations that means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement for expressions in logical proofs.

In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.

<span class="mw-page-title-main">Logical disjunction</span> Logical connective OR

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First-order logic—also called predicate logic, predicate calculus, quantificational logic—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantified variables over non-logical objects, and allows the use of sentences that contain variables. Rather than propositions such as "all men are mortal", in first-order logic one can have expressions in the form "for all x, if x is a man, then x is mortal"; where "for all x" is a quantifier, x is a variable, and "... is a man" and "... is mortal" are predicates. This distinguishes it from propositional logic, which does not use quantifiers or relations; in this sense, propositional logic is the foundation of first-order logic.

<span class="mw-page-title-main">Logical conjunction</span> Logical connective AND

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In logic, a logical connective is a logical constant. Connectives can be used to connect logical formulas. For instance in the syntax of propositional logic, the binary connective $can be used to join the two atomic formulas and, rendering the complex formula .$

In logic, mathematics, and computer science, arity is the number of arguments or operands taken by a function, operation or relation. In mathematics, arity may also be called rank, but this word can have many other meanings. In logic and philosophy, arity may also be called adicity and degree. In linguistics, it is usually named valency.

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<span class="mw-page-title-main">Exclusive or</span> True when either but not both inputs are true

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In mathematics, a Kleene algebra is an idempotent semiring endowed with a closure operator. It generalizes the operations known from regular expressions.

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In mathematics, a subset of a given set is closed under an operation of the larger set if performing that operation on members of the subset always produces a member of that subset. For example, the natural numbers are closed under addition, but not under subtraction: 1 − 2 is not a natural number, although both 1 and 2 are.

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In abstract algebra, a semiring is an algebraic structure. Semirings are a generalization of rings, dropping the requirement that each element must have an additive inverse. At the same time, semirings are a generalization of bounded distributive lattices.

In logic, a truth function is a function that accepts truth values as input and produces a unique truth value as output. In other words: the input and output of a truth function are all truth values; a truth function will always output exactly one truth value, and inputting the same truth value(s) will always output the same truth value. The typical example is in propositional logic, wherein a compound statement is constructed using individual statements connected by logical connectives; if the truth value of the compound statement is entirely determined by the truth value(s) of the constituent statement(s), the compound statement is called a truth function, and any logical connectives used are said to be truth functional.

In computer science, a relational operator is a programming language construct or operator that tests or defines some kind of relation between two entities. These include numerical equality and inequalities.

In computing, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another, called the modulus of the operation.

In logic, a functionally complete set of logical connectives or Boolean operators is one that can be used to express all possible truth tables by combining members of the set into a Boolean expression. A well-known complete set of connectives is { AND, NOT }. Each of the singleton sets { NAND } and { NOR } is functionally complete. However, the set { AND, OR } is incomplete, due to its inability to express NOT.

In abstract algebra, a semiheap is an algebraic structure consisting of a non-empty set H with a ternary operation denoted $that satisfies a modified associativity property:$

A truth table is a mathematical table used in logic—specifically in connection with Boolean algebra, Boolean functions, and propositional calculus—which sets out the functional values of logical expressions on each of their functional arguments, that is, for each combination of values taken by their logical variables. In particular, truth tables can be used to show whether a propositional expression is true for all legitimate input values, that is, logically valid.

References

1 2 MDN, nmve. "Conditional (ternary) Operator". Mozilla Developer Network. Retrieved 20 February 2017.
↑ Jeremiah Certaine (1943) The ternary operation (abc) = a b⁻¹c of a group, Bulletin of the American Mathematical Society 49: 868–77 MR 0009953
↑ 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
↑ Hoffer, Alex. "Ternary Operator". Cprogramming.com. Retrieved 20 February 2017.
↑ "6. Expressions — Python 3.9.1 documentation". docs.python.org. Retrieved 2021-01-19.
↑ "The OCaml Manual: Chapter 11 The OCaml language: (7) Expressions". ocaml.org. Retrieved 2023-05-03.

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[MDM_nmve-1] 1 2 MDN, nmve. "Conditional (ternary) Operator". Mozilla Developer Network. 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 MR 0009953

[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. Retrieved 20 February 2017.

[5] "6. Expressions — Python 3.9.1 documentation". docs.python.org. Retrieved 2021-01-19.

[6] "The OCaml Manual: Chapter 11 The OCaml language: (7) Expressions". ocaml.org. Retrieved 2023-05-03.

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