totally ordered set: meaning, definition, pronunciation and examples
C2Technical, Academic
Quick answer
What does “totally ordered set” mean?
A set of elements equipped with a relation that is transitive, antisymmetric, and total, meaning any two distinct elements are comparable.
Audio
Pronunciation
Definition
Meaning and Definition
A set of elements equipped with a relation that is transitive, antisymmetric, and total, meaning any two distinct elements are comparable.
A mathematical structure where every pair of elements can be ranked in a linear sequence without ties (except for equality). It's a fundamental concept in order theory and set theory, underlying the intuitive idea of a 'chain' of elements.
Dialectal Variation
British vs American Usage
Differences
No significant lexical or semantic differences. Spelling conventions follow standard British/American norms for surrounding text (e.g., 'ordered' vs. 'ordered').
Connotations
None beyond its precise mathematical definition.
Frequency
Used with identical frequency and meaning in mathematical contexts in both varieties.
Grammar
How to Use “totally ordered set” in a Sentence
The set X is a totally ordered set under the relation ≤.We consider the totally ordered set (S, <).Vocabulary
Collocations
Examples
Examples of “totally ordered set” in a Sentence
adjective
British English
- The set of real numbers with the usual less-than relation is totally ordered.
American English
- The rational numbers form a totally ordered set but are not well-ordered.
Usage
Meaning in Context
Business
Virtually never used.
Academic
Core terminology in mathematics, computer science (especially theory), and formal logic.
Everyday
Not used.
Technical
The primary domain of use. Essential in papers and textbooks on set theory, order theory, and discrete mathematics.
Vocabulary
Synonyms of “totally ordered set”
Strong
Neutral
Weak
Vocabulary
Antonyms of “totally ordered set”
Watch out
Common Mistakes When Using “totally ordered set”
- Confusing it with a 'well-ordered set' (which requires every non-empty subset to have a least element).
- Using it to mean simply 'sorted' or 'organized' in a non-mathematical context.
- Omitting the relation: saying 'the integers are a totally ordered set' instead of 'the integers under the standard ≤ relation form a totally ordered set.'
FAQ
Frequently Asked Questions
No. A well-ordered set is a special type of totally ordered set where every non-empty subset has a least element. The natural numbers are well-ordered, but the integers with the standard order are totally ordered but not well-ordered (the subset of negative integers has no least element).
The order relation is antisymmetric (if a ≤ b and b ≤ a, then a = b). So distinct elements must be comparable and one must be strictly less than the other. Equality is only for the same element.
In computer science, it's used in sorting algorithms (which require a total order on the data), database indexing, and in formal specification languages. In economics, it can model preferences under certain axioms.
People ranked by height (assuming no two people have exactly the same height) form a totally ordered set. If ties are allowed, it's technically a 'total preorder'.
Totally ordered set is usually technical, academic in register.
Totally ordered set: in British English it is pronounced /ˈtəʊtəli ˈɔːdəd ˌset/, and in American English it is pronounced /ˈtoʊtəli ˈɔːrdərd ˌset/. Tap the audio buttons above to hear it.
Learning
Memory Aids
Mnemonic
Think of a TOTALLY ORDERED SET as a TOTAL ranking—like a ladder (chain) where you can compare ANY two rungs to say which is higher.
Conceptual Metaphor
A LINE or CHAIN where every element has a definite place.
Practice
Quiz
Which of the following is NOT necessarily a totally ordered set?