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An unordered pair is a finite set; its cardinality (number of elements) is 2 or (if the two elements are not distinct) 1. In axiomatic set theory, the existence of unordered pairs is required by an axiom, the axiom of pairing. More generally, an unordered n-tuple is a set of the form {a 1, a 2,... a n}. [5] [6] [7]
add a new (,) pair to the collection, mapping the key to its new value. Any existing mapping is overwritten. The arguments to this operation are the key and the value. Remove or delete remove a (,) pair from the collection, unmapping a given key from its value. The argument to this operation is the key.
For example, to prove the case n = 3, use the axiom of pairing three times, to produce the pair {A 1,A 2}, the singleton {A 3}, and then the pair {{A 1,A 2},{A 3}}. The axiom of union then produces the desired result, {A 1,A 2,A 3}. We can extend this schema to include n=0 if we interpret that case as the axiom of empty set.
However, since the code only uses methods common to the interface Map, a self-balancing binary tree could be used by calling the constructor of the TreeMap class (which implements the subinterface SortedMap), without changing the definition of the phoneBook variable, or the rest of the code, or using other underlying data structures that ...
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: my %elements = map { $_ => 1 } @elements ; Other popular methods include arrays .
In computer programming, a collection is an abstract data type that is a grouping of items that can be used in a polymorphic way. Often, the items are of the same data type such as int or string. Sometimes the items derive from a common type; even deriving from the most general type of a programming language such as object or variant.
In many programming languages, map is a higher-order function that applies a given function to each element of a collection, e.g. a list or set, returning the results in a collection of the same type.
Representations might also be more complicated, for example using indexes or ancestor lists for performance. Trees as used in computing are similar to but can be different from mathematical constructs of trees in graph theory, trees in set theory, and trees in descriptive set theory.