N-ary ratio
In mathematics and logic, a n-ary relation R (or often commonly relation) is a generalization of the binary relation, where R is formed by a tuple of n terms:
- R={(x1,x2,...... ,xn):x1한 한 X1∧ ∧ x2한 한 X2∧ ∧ ...... ∧ ∧ xn한 한 Xn∧ ∧ R(x1,x2,...... ,xn)=verdaderor!{displaystyle R={(x_{1},x_{2},ldotsx_{n}):;x_{1}in X_{1}{1};x_{2}{2}{in X_{2}{2}{;ldots ;land ;xx1⁄2}{nx1⁄2}{l}{l}{l
A preaching n-ario: R(x1,x2,...... ,xn)=verdaderor{displaystyle R(x_{1},x_{2},ldotsx_{n})={rm {true}}}} is a function to true values of n variables.
Because a relationship like the previous one uniquely defines a preaching n-ario that is worth for x1,x2,...... ,xn{displaystyle x_{1},x_{2},ldotsx_{n} Yes and only if (x1,x2,...... ,xn){displaystyle (x_{1},x_{2},ldotsx_{n})} It's in. R{displaystyle R,}and vice versa, the relationship and the preached are often denoted with the same symbol. Thus, for example, the following two proposals are considered equivalent:
- R(x1,x2,...... ,xn){displaystyle R(x_{1},x_{2},ldotsx_{n}}}}}
- (x1,x2,...... ,xn)한 한 R{displaystyle (x_{1},x_{2},ldotsx_{n})in R}
Example
The following relation, defined on the set N of natural numbers, is n-ary, since it has n terms:
- <math alttext="{displaystyle C={(a_{1},a_{2},ldots;a_{n}):;(a_{1},a_{2},ldots;a_{n})in mathbb {N} ^{n};land ;(a_{1}<a_{2}<ldots ;C={(a1,a2,...... ,an):(a1,a2,...... ,an)한 한 Nn∧ ∧ (a1.a2....... .an)!{displaystyle C={(a_{1},a_{2},ldots;a_{n}):;(a_{1},a_{2},ldots;a_{n})in mathbb {N} ^{n};land ;(a_{1⁄2}{2}{s}{2}{n}{ldot}{ldot}}}}}{n}{<img alt="{displaystyle C={(a_{1},a_{2},ldots;a_{n}):;(a_{1},a_{2},ldots;a_{n})in mathbb {N} ^{n};land ;(a_{1}<a_{2}<ldots ;
The relation says that each of the terms is greater than the previous one. The value of n is a fixed parameter, which can be made explicit, or left as generic, to describe a general case.
Subtypes
Relations are classified according to the number of sets in the Cartesian product; in other words, the number of terms in the expression:
- Single relationship: R(x).
- Binary relationship: R(x, and).
- Alternative relationship: R(x, and, z).
- Quaternary relationship: R(x, and, z, t).
Relations with more than 4 terms are generally called n-ary; for example "a 5-ary relationship".