## Lower and upper

by Wei Wang

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## Lower and upper bounds

In mathematics, especially in order theory, an upper bound of a subset $S$ of some partially ordered set $K$ is an element of $K$ which is greater than or equal to every element of $S$. The term lower bound is defined dually as an element of K which is less than or equal to every element of $S$. A set with an upper bound is said to be bounded from above by that bound, a set with a lower bound is said to be bounded from below by that bound. The terms bounded above (bounded below) are also used in the mathematical literature for sets that have upper (respectively lower) bounds [1].
The definitions can be generalized to functions and even sets of functions. Given a function $f$ with domain $D$ and a partially ordered set $K$ as codomain, an element $y$ of $K$ is an upper bound of $f$ if $y\ge f(x)$ for each $x$ in $D$. The upper bound is called sharp if equality holds for at least one value of $x$. Function $g$ defined on domain $D$ and having the same codomain $K$ is an upper bound of $f$ if $g(x)\ge f(x)$ for each $x$ in $D$. Function $g$ is further said to be an upper bound of a set of functions if it is an upper bound of each function in that set. The notion of lower bound for (sets of) functions is defined analogously, with $\le$ replacing $\ge$.