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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.
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A subset of a poset is downward-directed if and only if its upper closure is a filter. Directed subsets are used in domain theory, which studies directed-complete partial orders. [5] These are posets in which every upward-directed set is required to have a least upper bound. In this context, directed subsets again provide a generalization of ...
The best known upper bound on the size of a square-difference-free set of numbers up to is only slightly sublinear, but the largest known sets of this form are significantly smaller, of size . Closing the gap between these upper and lower bounds remains an open problem .
A real number x is called an upper bound for S if x ≥ s for all s ∈ S. 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 ...
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"Upper bound" is correct in the first paragraph, where we speak of "accuracy". The Cramer-Rao inequality gives the maximum accuracy that can be achieved. Later, we speak about variance and there it is in fact a lower bound. High variance means low accuracy and vice versa.
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