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The ordered pair (a, b) is different from the ordered pair (b, a), unless a = b. In contrast, the unordered pair, denoted {a, b}, always equals the unordered pair {b, a}. Ordered pairs are also called 2-tuples, or sequences (sometimes, lists in a computer science context) of length 2. Ordered pairs of scalars are sometimes called 2-dimensional ...
In mathematics, an unordered pair or pair set is a set of the form {a, b}, i.e. a set having two elements a and b with no particular relation between them, where {a, b} = {b, a}. In contrast, an ordered pair (a, b) has a as its first element and b as its second element, which means (a, b) ≠ (b, a).
To guarantee a fixed order of enumeration, ordered versions of the associative array are often used. There are two senses of an ordered dictionary: The order of enumeration is always deterministic for a given set of keys by sorting. This is the case for tree-based implementations, one representative being the <map> container of C++. [16]
We can use the axiom of extensionality to show that this set C is unique. We call the set C the pair of A and B, and denote it {A,B}. Thus the essence of the axiom is: Any two objects have a pair. The set {A,A} is abbreviated {A}, called the singleton containing A. Note that a singleton is a special case of a pair.
similar to a set, multiset, map, or multimap, respectively, but implemented using a hash table; keys are not ordered, but a hash function must exist for the key type. These types were left out of the C++ standard; similar containers were standardized in C++11, but with different names (unordered_set and unordered_map). Other types of containers ...
Sometimes the items derive from a common type; even deriving from the most general type of a programming language such as object or variant. Although easily confused with implementations in programming languages, collection, as an abstract concept, refers to mathematical concepts which can be misunderstood when the focus is on an implementation.
r : E → {{x,y} : x, y ∈ V}, assigning to each edge an unordered pair of endpoint nodes. Some authors allow multigraphs to have loops , that is, an edge that connects a vertex to itself, [ 2 ] while others call these pseudographs , reserving the term multigraph for the case with no loops.
Ordered storage typically stores the records in order and may have to rearrange or increase the file size when a new record is inserted, resulting in lower insertion efficiency. However, ordered storage provides more efficient retrieval as the records are pre-sorted, resulting in a complexity of O ( log n ) {\displaystyle O\left(\log n ...