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Set (abstract data type) In computer science, a set is an abstract data type that can store unique values, without any particular order. It is a computer implementation of the mathematical concept of a finite set. Unlike most other collection types, rather than retrieving a specific element from a set, one typically tests a value for membership ...
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.
The total number of partitions of an n-element set is the Bell number B n. The first several Bell numbers are B 0 = 1, B 1 = 1, B 2 = 2, B 3 = 5, B 4 = 15, B 5 = 52, and B 6 = 203 (sequence A000110 in the OEIS). Bell numbers satisfy the recursion + = = and have the exponential generating function
Definition. The intersection of two sets and denoted by , [3] is the set of all objects that are members of both the sets and In symbols: That is, is an element of the intersection if and only if is both an element of and an element of [3] For example: The intersection of the sets {1, 2, 3} and {2, 3, 4} is {2, 3}.
In the Lehmer code for a permutation σ, the number d n represents the choice made for the first term σ 1, the number d n−1 represents the choice made for the second term σ 2 among the remaining n − 1 elements of the set, and so forth. More precisely, each d n+1−i gives the number of remaining elements strictly less than the term σ i.
In mathematics, specifically set theory, the Cartesian product of two sets A and B, denoted A × B, is the set of all ordered pairs (a, b) where a is in A and b is in B. [1] In terms of set-builder notation, that is [2][3] A table can be created by taking the Cartesian product of a set of rows and a set of columns.
One of the simplest and most natural examples is the multiset of prime factors of a natural number n. Here the underlying set of elements is the set of prime factors of n. For example, the number 120 has the prime factorization =, which gives the multiset {2, 2, 2, 3, 5}.
For a ≤ b, the closed interval [a, b] is the set of elements x satisfying a ≤ x ≤ b (that is, a ≤ x and x ≤ b). It contains at least the elements a and b. Using the corresponding strict relation "<", the open interval (a, b) is the set of elements x satisfying a < x < b (i.e. a < x and x < b). An open interval may be empty even if a < b.