ordered set
C2Technical / Academic / Formal
Definition
Meaning
A mathematical concept where elements are organized in a specific sequence, often with a defined binary relation.
A collection of items arranged according to a specific rule or hierarchy, where the order or position of elements is meaningful. The term can also be used metaphorically in non-technical contexts to describe any systematically arranged collection.
Linguistics
Semantic Notes
Primarily a mathematical term. It's a compound noun functioning as a singular unit, but 'ordered' can be inflected ('more ordered set'). In everyday use, it's often misinterpreted as simply 'a set that is tidy' rather than a technical sequence.
Dialectal Variation
British vs American Usage
Differences
No significant difference in meaning. Spelling conventions follow national standards for the components ('ordered' vs 'ordered', 'set' vs 'set'). The hyphenated form 'ordered-set' is rarely, if ever, used in either variety.
Connotations
Identical technical connotations. Potential informal misinterpretation ('a set that has been put in order') is equally likely in both varieties.
Frequency
Extremely rare in general discourse. Its frequency is confined almost exclusively to academic and technical writing (mathematics, computer science, logic), with identical distribution between UK and US English in those domains.
Vocabulary
Collocations
Grammar
Valency Patterns
[Verb] an ordered set: define, construct, consider, analyze, denoteAn ordered set of [Noun Phrase]: an ordered set of numbers, an ordered set of rules, an ordered set of tasks[Adjective] ordered set: partially, totally, linearly, well-Vocabulary
Synonyms
Strong
Neutral
Weak
Vocabulary
Antonyms
Phrases
Idioms & Phrases
- “In short supply”
Usage
Context Usage
Business
Virtually never used. Might appear metaphorically in highly formal strategy documents, e.g., 'an ordered set of strategic priorities.'
Academic
The primary domain. Used in mathematics, computer science, logic, and formal linguistics to describe structured collections with defined relations (e.g., 'A lattice is a specific type of ordered set.').
Everyday
Extremely rare and likely to be misunderstood. If used, it would be non-technical, e.g., 'She presented an ordered set of arguments.'
Technical
Core terminology in mathematics and theoretical computer science. Precision is paramount, distinguishing between types like 'partially ordered set' (poset) and 'totally ordered set'.
Examples
By Part of Speech
verb
British English
- We can order the set of natural numbers by magnitude.
- The algorithm first orders the set of vertices.
American English
- We can order the set of real numbers by value.
- The program orders the data set chronologically.
adverb
British English
- The elements were arranged orderedly within the set.
- The tasks proceeded in an orderedly set sequence.
American English
- The files are stored orderedly in the dataset.
- The list was orderedly set by date.
adjective
British English
- The data structure requires a more ordered set of inputs.
- They worked with a partially-ordered set.
American English
- The system functions best with a more ordered set of parameters.
- This is a totally-ordered set.
Examples
By CEFR Level
- The teacher asked for an ordered set of coloured pencils, from light to dark.
- For the experiment, you need an ordered set of test tubes, numbered one to ten.
- In mathematics, an ordered set is fundamental to understanding concepts like sequences and inequalities.
- The proof relies on constructing a well-ordered set that satisfies Zorn's lemma, thereby establishing a maximal element.
Learning
Memory Aids
Mnemonic
Think of a **SET** of encyclopedias on a shelf. They are in an ORDER (A, B, C...). An ORDERED SET is like that—a collection where the sequence (the order) is a defined part of what it is.
Conceptual Metaphor
HIERARCHY IS ORDER (e.g., a company org chart as an ordered set of positions). SEQUENCE IS A PATH (e.g., steps in a recipe as an ordered set of instructions).
Watch out
Common Pitfalls
Translation Traps (for Russian speakers)
- Не переводите как 'упорядоченный набор' в бытовом смысле ('аккуратный набор'). В техническом контексте это корректно, но в математике стандартный термин — 'упорядоченное множество' или 'частично упорядоченное множество'.
- Избегайте кальки 'заказанный сет' — это грубая ошибка.
- Вне математики лучше использовать 'последовательность' или 'упорядоченный список'.
Common Mistakes
- Using 'ordered set' to mean 'a set that has been tidied up'. (Incorrect: 'Please put the cutlery in an ordered set.')
- Treating it as a plural noun. (Incorrect: 'These ordered sets are...' is fine, but 'An ordered set are...' is wrong.)
- Confusing 'ordered set' with 'set order' (which is a request for a sequence or a fixed arrangement).
Practice
Quiz
In which field is the term 'ordered set' most precisely and frequently used?
FAQ
Frequently Asked Questions
Very closely related. A sequence is often defined as a function from the natural numbers, implying a total order. An 'ordered set' is a more general term; a sequence can be seen as a manifestation of a totally ordered set.
It's not recommended for general communication due to its strong technical association and high potential for misunderstanding. Use 'sequence', 'list in order', or 'arranged collection' instead.
In standard set theory, a set has no inherent order ({1,2} = {2,1}). An 'ordered set' includes order as an essential, defining property, making <1,2> different from <2,1>.
No, that is a common confusion. In 'ordered set', 'ordered' is an adjective meaning 'arranged systematically'. In 'ordered a pizza', it's a verb meaning 'requested'. They are different lexemes.