Search results
Results From The WOW.Com Content Network
3.5.1.1 Empty set. 3.5.2 Meets, Joins, ... 6.1 Binary ⨯ distributes over ... Sets that do not intersect are said to be disjoint.
More generally, one may define upper bound and least upper bound for any subset of a partially ordered set X, with “real number” replaced by “element of X ”. In this case, we say that X has the least-upper-bound property if every non-empty subset of X with an upper bound has a least upper bound in X.
[5] [6] Since relations are sets, they can be manipulated using set operations, including union, intersection, and complementation, leading to the algebra of sets. Furthermore, the calculus of relations includes the operations of taking the converse and composing relations. [7] [8] [9]
Set-builder notation can be used to describe a set that is defined by a predicate, that is, a logical formula that evaluates to true for an element of the set, and false otherwise. [2]
In particular, totally ordered sets can also be referred to as "ordered sets", especially in areas where these structures are more common than posets. Some authors use different symbols than ≤ {\displaystyle \leq } such as ⊑ {\displaystyle \sqsubseteq } [ 6 ] or ⪯ {\displaystyle \preceq } [ 7 ] to distinguish partial orders from total orders.
The most general notion is the union of an arbitrary collection of sets, sometimes called an infinitary union. If M is a set or class whose elements are sets, then x is an element of the union of M if and only if there is at least one element A of M such that x is an element of A. [11] In symbols:
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 ...
The intersection of all the convex sets that contain a given subset A of Euclidean space is called the convex hull of A. It is the smallest convex set containing A . A convex function is a real-valued function defined on an interval with the property that its epigraph (the set of points on or above the graph of the function) is a convex set.