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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.
In probability theory, Popoviciu's inequality, named after Tiberiu Popoviciu, is an upper bound on the variance σ 2 of any bounded probability distribution.Let M and m be upper and lower bounds on the values of any random variable with a particular probability distribution.
In mathematics, the limit of a sequence of sets,, … (subsets of a common set ) is a set whose elements are determined by the sequence in either of two equivalent ways: (1) by upper and lower bounds on the sequence that converge monotonically to the same set (analogous to convergence of real-valued sequences) and (2) by convergence of a sequence of indicator functions which are themselves ...
A real number x is the least upper bound (or supremum) for S if x is an upper bound for S and x ≤ y for every upper bound y of S. The least-upper-bound property states that any non-empty set of real numbers that has an upper bound must have a least upper bound in real numbers .
The feasible regions of linear programming are defined by a set of inequalities.. In mathematics, an inequality is a relation which makes a non-equal comparison between two numbers or other mathematical expressions. [1]
As shown in the image, lower and upper bounds for the circumference of a circle can be obtained with inscribed and circumscribed regular polygons. When examining a circle with diameter 1 {\displaystyle 1} , the circumference is (by definition of Pi) the circle number π {\displaystyle \pi } .
Navarro [17] proved that these bounds are sharp, that is, they are the best possible bounds for that regions when we just know the mean and the covariance matrix of X. Stellato et al. [ 18 ] showed that this multivariate version of the Chebyshev inequality can be easily derived analytically as a special case of Vandenberghe et al. [ 19 ] where ...
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) .
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