Search results
Results From The WOW.Com Content Network
8.3 (Pre)Images of binary set operations 8.3.1 Counter-examples: images of operations not distributing 8.3.2 Conditions guaranteeing that images distribute over set operations
As an example, "is less than" is a relation on the set of natural numbers; it holds, for instance, between the values 1 and 3 (denoted as 1 < 3), and likewise between 3 and 4 (denoted as 3 < 4), but not between the values 3 and 1 nor between 4 and 4, that is, 3 < 1 and 4 < 4 both evaluate to false.
The open sets in the product topology are unions (finite or infinite) of sets of the form , where each U i is open in X i and U i ≠ X i only finitely many times. In particular, for a finite product (in particular, for the product of two topological spaces), the products of base elements of the X i gives a basis for the product ∏ i ∈ I X i ...
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 set theory and related branches of mathematics, the von Neumann universe, or von Neumann hierarchy of sets, denoted by V, is the class of hereditary well-founded sets. This collection, which is formalized by Zermelo–Fraenkel set theory (ZFC), is often used to provide an interpretation or motivation of the axioms of ZFC.
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: x ∈ ⋃ M ∃ A ∈ M , x ∈ A . {\displaystyle x\in \bigcup \mathbf {M} \iff \exists A\in \mathbf {M} ,\ x\in A.}
The convex-hull operation is needed for the set of convex sets to form a lattice, in which the "join" operation is the convex hull of the union of two convex sets = = ( ()). The intersection of any collection of convex sets is itself convex, so the convex subsets of a (real or complex) vector space form a complete lattice .
An infinite sequence of real numbers (in blue). This sequence is neither increasing, decreasing, convergent, nor Cauchy.It is, however, bounded. In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters.