Search results
Results From The WOW.Com Content Network
If an infinite set is partitioned into finitely many subsets, then at least one of them must be infinite. Any set which can be mapped onto an infinite set is infinite. The Cartesian product of an infinite set and a nonempty set is infinite. The Cartesian product of an infinite number of sets, each containing at least two elements, is either ...
[1] [3] For example, if a line is viewed as the set of all of its points, their infinite number (i.e., the cardinality of the line) is larger than the number of integers. [4] In this usage, infinity is a mathematical concept, and infinite mathematical objects can be studied, manipulated, and used just like any other mathematical object.
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.
An infinite series of any rational function of can be reduced to a finite series of polygamma functions, by use of partial fraction decomposition, [8] as explained here. This fact can also be applied to finite series of rational functions, allowing the result to be computed in constant time even when the series contains a large number of terms.
the set of all finite subsets of any given countably infinite set. Among the countably infinite sets are certain infinite ordinals, [c] including for example ω, ω + 1, ω⋅2, ω 2, ω ω, and ε 0. [6] For example, the sequence (with ordinality ω⋅2) of all positive odd integers followed by all positive even integers
The best known example of an uncountable set is the set of all real numbers; Cantor's diagonal argument shows that this set is uncountable. The diagonalization proof technique can also be used to show that several other sets are uncountable, such as the set of all infinite sequences of natural numbers (see: (sequence A102288 in the OEIS)), and the set of all subsets of the set ...
For example, (aleph-nought) denotes the smallest infinite cardinal number (representing the size of the set of natural numbers), and denotes the smallest infinite ordinal number. The infinity symbol may also be used to represent a point at infinity , especially when there is only one such point under consideration.
In mathematics, transfinite numbers or infinite numbers are numbers that are "infinite" in the sense that they are larger than all finite numbers. These include the transfinite cardinals, which are cardinal numbers used to quantify the size of infinite sets, and the transfinite ordinals, which are ordinal numbers used to provide an ordering of infinite sets.