Infimum and supremum
In mathematics, the infimum or greatest lower bound of a set A, written as [math]\displaystyle{ \inf(A) }[/math], is the greatest element among all lower bounds of A. Similarly, the supremum or least upper bound of A, written as [math]\displaystyle{ \sup(A) }[/math], is the smallest element among all upper bounds of A.[1]
For example, if A is the set [math]\displaystyle{ \{ \tfrac{1}{1}, \tfrac{1}{2}, \tfrac{1}{3}, \ldots \} }[/math], then [math]\displaystyle{ \inf(A) = 0 }[/math] and [math]\displaystyle{ \sup(A) = 1 }[/math]. Infimum and supremum are unique, if they exist.[2][3] Infimum and supremum are key concepts in the field of mathematical analysis.
Infimum And Supremum Media
A set P of real numbers (hollow and filled circles), a subset S of P (filled circles), and the infimum of S. Note that for totally ordered finite sets, the infimum and the minimum are equal.
A set A of real numbers (blue circles), a set of upper bounds of A (red diamond and circles), and the smallest such upper bound, that is, the supremum of A (red diamond).
supremum = least upper bound
Related pages
References
- ↑ "List of Calculus and Analysis Symbols". Math Vault. 2020-05-11. Retrieved 2020-10-14.
- ↑ Weisstein, Eric W. "Supremum". mathworld.wolfram.com. Retrieved 2020-10-14.
- ↑ Weisstein, Eric W. "Infimum". mathworld.wolfram.com. Retrieved 2020-10-14.