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The union of two intervals is an interval if and only if they have a non-empty intersection or an open end-point of one interval is a closed end-point of the other, for example (,) [,] = (,]. If R {\displaystyle \mathbb {R} } is viewed as a metric space , its open balls are the open bounded intervals ( c + r , c − r ) , and its closed balls ...
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.
Instead, the inequalities must be solved independently, yielding x < 1 / 2 and x ≥ −1 respectively, which can be combined into the final solution −1 ≤ x < 1 / 2 . Occasionally, chained notation is used with inequalities in different directions, in which case the meaning is the logical conjunction of the inequalities ...
3. In set-builder notation, it is used as a separator meaning "such that"; see { : }. / 1. Denotes division and is read as divided by or over. Often replaced by a horizontal bar. For example, 3 / 2 or . 2. Denotes a quotient structure.
The notation is also used to denote the characteristic function in convex analysis, which is defined as if using the reciprocal of the standard definition of the indicator function. A related concept in statistics is that of a dummy variable .
This partitions the interval [a, b] into n sub-intervals [x i−1, x i] indexed by i, each of which is "tagged" with a specific point t i ∈ [x i−1, x i]. A Riemann sum of a function f with respect to such a tagged partition is defined as
Infix notation is a method similar to immediate execution with AESH and/or AESP, but unary operations are input into the calculator in the same order as they are written on paper. Calculators that use infix notation tend to incorporate a dot-matrix display to display the expression being entered, frequently accompanied by a seven-segment ...
Hardy's inequality is an inequality in mathematics, named after G. H. Hardy. Its discrete version states that if a 1 , a 2 , a 3 , … {\displaystyle a_{1},a_{2},a_{3},\dots } is a sequence of non-negative real numbers , then for every real number p > 1 one has