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The main objective of interval arithmetic is to provide a simple way of calculating upper and lower bounds of a function's range in one or more variables. These endpoints are not necessarily the true supremum or infimum of a range since the precise calculation of those values can be difficult or impossible; the bounds only need to contain the function's range as a subset.
13934 and other numbers x such that x ≥ 13934 would be an upper bound for S. 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 ...
of a Riemann integrable function defined on a closed and bounded interval are the real numbers and , in which is called the lower limit and the upper limit. The region that is bounded can be seen as the area inside a {\displaystyle a} and b {\displaystyle b} .
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 } .
If () = ([,]) (that is, the supremum of over [,]), the method is the upper rule and gives an upper Riemann sum or upper Darboux sum. If f ( x i ∗ ) = inf f ( [ x i − 1 , x i ] ) {\displaystyle f(x_{i}^{*})=\inf f([x_{i-1},x_{i}])} (that is, the infimum of f over [ x i − 1 , x i ] {\displaystyle [x_{i-1},x_{i}]} ), the method is the lower ...
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 ...
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The digits in the "odd place-intervals", i.e. between digits 2 2n+1 and 2 2n+2 − 1 are not restricted and may take any value. This fractal has upper box dimension 2/3 and lower box dimension 1/3, a fact which may be easily verified by calculating N(ε) for = and noting that their values behave differently for n even and odd.