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The set S = {42} has 42 as both an upper bound and a lower bound; all other numbers are either an upper bound or a lower bound for that S. Every subset of the natural numbers has a lower bound since the natural numbers have a least element (0 or 1, depending on convention). An infinite subset of the natural numbers cannot be bounded from above.
Boole's inequality may be generalized to find upper and lower bounds on the probability of finite unions of events. [2] These bounds are known as Bonferroni inequalities , after Carlo Emilio Bonferroni ; see Bonferroni (1936) .
There is a corresponding greatest-lower-bound property; an ordered set possesses the greatest-lower-bound property if and only if it also possesses the least-upper-bound property; the least-upper-bound of the set of lower bounds of a set is the greatest-lower-bound, and the greatest-lower-bound of the set of upper bounds of a set is the least ...
Toggle Upper and lower bounds subsection. 2.1 General placement methods. 2.2 Upper bound. 2.3 Conjectured bounds. ... upper bound is known. [19] Higher dimensions ...
That is to say, each successive change in the bounds of the interval within which must lie allows the value of to be estimated with a greater precision, either by increasing the lower bounds of the interval or decreasing the upper bounds of the interval.
The element k is called an upper bound of S. The concepts of bounded below and lower bound are defined similarly. (See also upper and lower bounds.) A subset S of a partially ordered set P is called bounded if it has both an upper and a lower bound, or equivalently, if it is contained in an interval.
Thus, the infimum or meet of a collection of subsets is the greatest lower bound while the supremum or join is the least upper bound. In this context, the inner limit, lim inf X n, is the largest meeting of tails of the sequence, and the outer limit, lim sup X n, is the smallest joining of tails of the sequence. The following makes this precise.
The bound combines the level with the average value of . In probability theory, Markov's inequality gives an upper bound on the probability that a non-negative random variable is greater than or equal to some positive constant. Markov's inequality is tight in the sense that for each chosen positive constant, there exists a random variable such ...