Search results
Results from the WOW.Com Content Network
In ZF, a set is infinite if and only if the power set of its power set is a Dedekind-infinite set, having a proper subset equinumerous to itself. [4] If the axiom of choice is also true, then infinite sets are precisely the Dedekind-infinite sets. If an infinite set is a well-orderable set, then it has many well-orderings which are non-isomorphic.
is a finite set with five elements. The number of elements of a finite set is a natural number (possibly zero) and is called the cardinality (or the cardinal number) of the set. A set that is not a finite set is called an infinite set. For example, the set of all positive integers is infinite:
A set of polygons in an Euler diagram This set equals the one depicted above since both have the very same elements.. In mathematics, a set is a collection of different [1] things; [2] [3] [4] these things are called elements or members of the set and are typically mathematical objects of any kind: numbers, symbols, points in space, lines, other geometrical shapes, variables, or even other ...
3. A transitive set or class is a set or class such that the membership relation is transitive on it. 4. A transitive model is a model of set theory that is transitive and has the usual membership relation tree 1. A tree is a partially ordered set (T, <) such that for each t ∈ T, the set {s ∈ T : s < t} is well-ordered by the relation < 2.
Examples of ICC groups are the group of permutations of an infinite set that leave all but a finite subset of elements fixed, [1]: 908 and free groups on two generators. [1]: 908 In abelian groups, every conjugacy class consists of only one element, so ICC groups are, in a way, as far from being abelian as possible.
2D-plot: As a generalization of a Boolean matrix, a relation on the –infinite– set R of real numbers can be represented as a two-dimensional geometric figure: using Cartesian coordinates, draw a point at (x,y) whenever (x,y) ∈ R. A transitive [c] relation R on a finite set X may be also represented as
Infinite sets are so common, that when one considers finite sets, this is generally explicitly stated; for example finite geometry, finite field, etc. Fermat's Last Theorem is a theorem that was stated in terms of elementary arithmetic , which has been proved only more than 350 years later.
The relevant section of Two New Sciences is excerpted below: [2]. Simplicio: Here a difficulty presents itself which appears to me insoluble.Since it is clear that we may have one line greater than another, each containing an infinite number of points, we are forced to admit that, within one and the same class, we may have something greater than infinity, because the infinity of points in the ...