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infimum

Very Low
UK/ˈɪnfɪməm/US/ˈɪnfɪməm/

Technical / Academic

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Definition

Meaning

In mathematics, the greatest element that is less than or equal to all elements of a given set.

A term from order theory and real analysis denoting the greatest lower bound of a subset of a partially ordered set. In a set of real numbers, it is the largest number that is not greater than any number in the set.

Linguistics

Semantic Notes

The concept is dual to 'supremum' (least upper bound). For a set that has a minimum element, the infimum and minimum are the same. The term is primarily used in pure and applied mathematics, computer science (especially lattice theory), and formal logic.

Dialectal Variation

British vs American Usage

Differences

No significant differences in meaning or usage. Spelling and pronunciation are identical.

Connotations

Purely technical with no regional connotations.

Frequency

Extremely rare in general discourse in both regions, confined to specialised academic and technical contexts.

Vocabulary

Collocations

strong
greatest lower boundsetsubsetreal numberssupremum
medium
calculate thefind thedefinition ofproperty ofexists
weak
functionsequenceboundedclosedinterval

Grammar

Valency Patterns

The infimum of [SET]to find/take/calculate the infimuminfimum exists/is equal to

Vocabulary

Synonyms

Neutral

greatest lower bound

Weak

lower bound (less precise)

Vocabulary

Antonyms

supremumleast upper bound

Usage

Context Usage

Business

Virtually never used.

Academic

Exclusively used in mathematics, computer science, and related theoretical disciplines.

Everyday

Not used.

Technical

Core term in order theory, real analysis, optimisation, and lattice theory.

Examples

By CEFR Level

B2
  • The infimum of the set {1, 1/2, 1/3, ...} is 0.
  • To solve the problem, we first need to find the infimum of the data set.
C1
  • The infimum of a family of functions is defined pointwise.
  • If the set is not bounded below, its infimum is considered to be negative infinity.
  • Proving the existence of the infimum is a crucial step in the lemma.

Learning

Memory Aids

Mnemonic

Think 'IN-FIM-um': IN a set, it's the FIMal (final) point at the bottom—the greatest of the lower bounds.

Conceptual Metaphor

THE BOTTOM LINE (in a precise, mathematical sense).

Watch out

Common Pitfalls

Translation Traps (for Russian speakers)

  • Do not confuse with 'нижняя грань' (which is correct). Avoid literal translations like 'инфимум' unless in a strict mathematical context; the term is a direct Latin borrowing.

Common Mistakes

  • Using 'infimum' to mean 'minimum' (the minimum must be an element of the set, the infimum need not be).
  • Pronouncing it as /ɪnˈfaɪməm/.
  • Using it in non-mathematical contexts.

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 infimum and the minimum of a set?

FAQ

Frequently Asked Questions

No. The minimum must be an element of the set. The infimum is the greatest number that is less than or equal to all elements of the set, but it does not have to be a member of the set.

Almost exclusively in mathematics (especially real analysis, order theory, optimisation) and theoretical computer science (e.g., lattice theory in formal methods).

The standard plural is 'infima', following its Latin origin. 'Infimums' is also occasionally seen but is less traditional.

The direct opposite is 'supremum' or 'least upper bound'. While 'maximum' is the opposite of 'minimum', it is not the precise dual of 'infimum'.