ordered set
An "ordered set" is a fundamental concept in mathematics used to describe collections of elements with specific relational properties.
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Definition
C1Mathematics
(technical, academic)A collection of elements with a defined sequence based on a relation that is reflexive, antisymmetric, and transitive.
Example
- A set of integers with the usual less-than-or-equal relation forms an ordered set.
- The hierarchy of job positions in a company can be modeled as an ordered set.
C2Mathematics
(technical, academic)A collection where every pair of elements is comparable, meaning one element is either less than, equal to, or greater than the other.
Example
- The real numbers with the usual less-than relation form a totally ordered set.
- In a chain of command, every position is comparable to every other position, creating a totally ordered set.
C2Mathematics
(technical, academic)A set in which every non-empty subset has a least element.
Example
- The natural numbers with the usual less-than relation form a well-ordered set.
- In a well-ordered set, you can always find a smallest element in any subset.
Similar
Terms that have similar or relatively close meanings to "ordered set":
partially ordered setordered ringin orderitem setrunning orderin sortlexicographic orderback orderdata setopen setordinal numberunderlying set