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Infinite Set
In set theory, an infinite set is a set that is not a finite set. Infinite sets may be countable or uncountable. Properties The set of natural numbers (whose existence is postulated by the axiom of infinity) is infinite. It is the only set that is directly required by the axioms to be infinite. The existence of any other infinite set can be proved in Zermelo–Fraenkel set theory (ZFC), but only by showing that it follows from the existence of the natural numbers. A set is infinite if and only if for every natural number, the set has a subset whose cardinality is that natural number. If the axiom of choice holds, then a set is infinite if and only if it includes a countable infinite subset. If a set of sets is infinite or contains an infinite element, then its union is infinite. The power set of an infinite set is infinite. Any superset of an infinite set is infinite. If an infinite set is partitioned into finitely many subsets, then at least one of them must be infinite. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cardinality
In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized to infinite sets, which allows one to distinguish between different types of infinity, and to perform arithmetic on them. There are two approaches to cardinality: one which compares sets directly using bijections and injections, and another which uses cardinal numbers. The cardinality of a set is also called its size, when no confusion with other notions of size is possible. The cardinality of a set A is usually denoted , A, , with a vertical bar on each side; this is the same notation as absolute value, and the meaning depends on context. The cardinality of a set A may alternatively be denoted by n(A), , \operatorname(A), or \#A. History A crude sense of cardinality, an awareness that groups of things or events compare with other grou ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
