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That is, the power set ℘ of a finite set S is finite, with cardinality | |. Any subset of a finite set is finite. The set of values of a function when applied to elements of a finite set is finite. All finite sets are countable, but not all countable sets are finite. (Some authors, however, use "countable" to mean "countably infinite", so do ...
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 sets. [5] A set may have a finite number of elements or be an infinite set.
In mathematics, a cardinal number, or cardinal for short, is what is commonly called the number of elements of a set. In the case of a finite set, its cardinal number, or cardinality is therefore a natural number.
Each set of elements has a least upper bound (their "join") and a greatest lower bound (their "meet"), so that it forms a lattice, and more specifically (for partitions of a finite set) it is a geometric and supersolvable lattice. [6] [7] The partition lattice of a 4-element set has 15 elements and is depicted in the Hasse diagram on the left.
In mathematics, an element (or member ... also called set membership, is denoted by the symbol "∈". Writing ... while a finite set is a set with a finite number of ...
Note: The empty set symbol ∅ looks similar, but is unrelated to the Greek letter. or represents: the golden ratio 1.618... in mathematics, art, and architecture; Euler's totient function in number theory; the argument of a complex number in mathematics; the value of a plane angle in physics and mathematics
In mathematics, particularly in set theory, the aleph numbers are a sequence of numbers used to represent the cardinality (or size) of infinite sets that can be well-ordered. They were introduced by the mathematician Georg Cantor [ 1 ] and are named after the symbol he used to denote them, the Hebrew letter aleph (ℵ).
This article lists mathematical properties and laws of sets, involving the set-theoretic operations of union, intersection, and complementation and the relations of set equality and set inclusion. It also provides systematic procedures for evaluating expressions, and performing calculations, involving these operations and relations.