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supremum: meaning, definition, pronunciation and examples

C2
UK/suːˈpriːməm/US/sʊˈpriməm/

Formal, Academic, Technical

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Quick answer

What does “supremum” mean?

The least upper bound of a set.

Audio

Pronunciation

Definition

Meaning and Definition

The least upper bound of a set; the smallest element that is greater than or equal to every element in the set.

In mathematics, the supremum is a fundamental concept in real analysis and order theory, representing the 'maximum' of a set when a true maximum does not exist. In broader contexts, it can metaphorically refer to the highest attainable point or ultimate limit of something.

Dialectal Variation

British vs American Usage

Differences

No significant difference in meaning or usage. The term is international in mathematics.

Connotations

Purely technical; carries no cultural or regional connotations.

Frequency

Equally rare and specialised in both varieties, confined to university-level mathematics.

Grammar

How to Use “supremum” in a Sentence

the supremum of [set]the supremum over [range]find the supremum for [function]

Vocabulary

Collocations

strong
least upper boundessential supremumsupremum normsupremum metric
medium
calculate the supremumsupremum existsfinite supremumsupremum property
weak
absolute supremumglobal supremumsupremum value

Examples

Examples of “supremum” in a Sentence

verb

British English

  • The function is then supremised over the interval.

American English

  • We need to supremize the cost function.

adjective

British English

  • The supremum value was approached but never attained.

American English

  • The supremum norm is crucial in this analysis.

Usage

Meaning in Context

Business

Virtually never used.

Academic

Core concept in real analysis, measure theory, and functional analysis.

Everyday

Extremely rare; would only be used by someone discussing advanced mathematics.

Technical

Precise term in mathematics, computer science (e.g., in formal verification), and theoretical economics.

Vocabulary

Synonyms of “supremum”

Strong

join (in lattice theory)

Weak

upper limitmaximum (when it exists)

Vocabulary

Antonyms of “supremum”

infimumgreatest lower boundmeet

Watch out

Common Mistakes When Using “supremum”

  • Using 'supremum' to mean 'maximum' in casual speech.
  • Pronouncing it as /ˈsjuːprəməm/.
  • Misspelling as 'supremium'.
  • Using it without defining the relevant set.

FAQ

Frequently Asked Questions

No. The supremum is the least upper bound. It is a member of the set if and only if the set has a maximum element.

The maximum must be an element of the set. The supremum is the smallest number that is greater than or equal to all elements of the set; it does not have to be in the set.

Yes. If a set of real numbers is not bounded above, its supremum is defined to be +∞.

It is a foundational concept in real analysis, measure theory, probability, and any field relying on the formal properties of the real number system.

The least upper bound of a set.

Supremum is usually formal, academic, technical in register.

Supremum: in British English it is pronounced /suːˈpriːməm/, and in American English it is pronounced /sʊˈpriməm/. Tap the audio buttons above to hear it.

Phrases

Idioms & Phrases

  • “to take the supremum over”
  • “supremum and infimum”

Learning

Memory Aids

Mnemonic

Think 'SUPreme' + 'UM' – the 'most supreme' (highest) number that still bounds the set from above.

Conceptual Metaphor

THE ULTIMATE CEILING; THE FINAL BOSS OF UPPER BOUNDS.

Practice

Quiz

Fill in the gap
For the set S = {x ∈ ℝ : x < 2}, the is 2, even though 2 is not in S.
Multiple Choice

What is the relationship between the maximum and the supremum of a set?

supremum: meaning, definition, pronunciation and examples | Lingvocore