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List comprehension is a syntactic construct available in some programming languages for creating a list based on existing lists. It follows the form of the mathematical set-builder notation (set comprehension) as distinct from the use of map and filter functions.
Here, the list [0..] represents , x^2>3 represents the predicate, and 2*x represents the output expression.. List comprehensions give results in a defined order (unlike the members of sets); and list comprehensions may generate the members of a list in order, rather than produce the entirety of the list thus allowing, for example, the previous Haskell definition of the members of an infinite list.
A sorted linear hash table [8] may be used to provide deterministically ordered sets. Further, in languages that support maps but not sets, sets can be implemented in terms of maps. For example, a common programming idiom in Perl that converts an array to a hash whose values are the sentinel value 1, for use as a set, is:
6 + 2; 5 + 3; 5 + 2 + 1; 4 + 3 + 1; This is a general property. For each positive number, the number of partitions with odd parts equals the number of partitions with distinct parts, denoted by q(n). [8] [9] This result was proved by Leonhard Euler in 1748 [10] and later was generalized as Glaisher's theorem.
''Title of list:'' example 1, example 2, example 3 Title of list: example 1, example 2, example 3 This style requires less space on the page, and is preferred if there are only a few entries in the list, it can be read easily, and a direct edit point is not required. The list items should start with a lowercase letter unless they are proper nouns.
An example of this is R 3 = R × R × R, with R again the set of real numbers, [1] and more generally R n. The n-ary Cartesian power of a set X is isomorphic to the space of functions from an n-element set to X. As a special case, the 0-ary Cartesian power of X may be taken to be a singleton set, corresponding to the empty function with codomain X.
The heapsort algorithm can be divided into two phases: heap construction, and heap extraction. The heap is an implicit data structure which takes no space beyond the array of objects to be sorted; the array is interpreted as a complete binary tree where each array element is a node and each node's parent and child links are defined by simple arithmetic on the array indexes.
A set is countable if it can be enumerated, that is, if there exists an enumeration of it. Otherwise, it is uncountable. For example, the set of the real numbers is uncountable. A set is finite if it can be enumerated by means of a proper initial segment {1, ..., n} of the natural numbers, in which case, its cardinality is n.