Infimum and supremum

Infimum and Supremum are fundamental concepts in the field of mathematics, particularly within the areas of order theory and real analysis. These concepts are used to describe the greatest lower bound (infimum) and the least upper bound (supremum) of a set, respectively. Understanding these concepts is crucial for the study of sequences, functions, and various mathematical structures.

Definition

Given a set S in a partially ordered set P, the infimum (denoted as inf(S)) is the greatest element in P that is less than or equal to all elements of S. Conversely, the supremum (denoted as sup(S)) is the least element in P that is greater than or equal to all elements of S. If S is a subset of the real numbers (R), these definitions are often intuitively understood as the "greatest lower bound" and "least upper bound" of S, respectively.

Existence

The existence of an infimum or supremum of a set S depends on the properties of the partially ordered set P. In the case of the real numbers, the Completeness Axiom guarantees that every non-empty set of real numbers that is bounded below has an infimum and every non-empty set of real numbers that is bounded above has a supremum in R.

Properties

1. Uniqueness: If an infimum or supremum exists, it is unique. 2. Boundedness: If a set S has an infimum or supremum, then S is bounded below or above, respectively. 3. Order Preservation: If S ⊆ T and both infimum and supremum exist for these sets, then inf(T) ≤ inf(S) and sup(S) ≤ sup(T).

Examples

1. For the set of real numbers S = {x ∈ R | x < 2}, the supremum is 2, even though 2 is not an element of S. 2. The set of all negative real numbers has an infimum of -∞ in the extended real number system.

Applications

Infimum and supremum are used in various mathematical disciplines: - In calculus, they are used to define limits and continuity. - In measure theory, they play a role in the definition of Lebesgue integration. - In optimization, problems often involve finding the infimum or supremum of a function over a certain domain.

See Also

• Complete lattice
• Upper and lower bounds
• Limit (mathematics)
• Continuity