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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.
Equivalently, it stores a partition of a set into disjoint subsets. It provides operations for adding new sets, merging sets (replacing them with their union), and finding a representative member of a set. The last operation makes it possible to determine efficiently whether any two elements belong to the same set or to different sets.
A spreadsheet's concatenate ("&") function is used to assemble a complex text string—in this example, XML code for an SVG "circle" element. In formal language theory and computer programming, string concatenation is the operation of joining character strings end-to-end. For example, the concatenation of "snow" and "ball" is "snowball".
A set whose elements can be put into a one-to-one correspondence with the set of natural numbers, making it countable. enumeration The process of listing or counting elements in a set, especially for countable sets. epsilon 1. An epsilon number is an ordinal α such that α=ω α 2. Epsilon zero (ε 0) is the smallest epsilon number equinumerous
Church was evidently unaware that string theory already had two axiomatizations from the 1930s: one by Hans Hermes and one by Alfred Tarski. [3] Coincidentally, the first English presentation of Tarski's 1933 axiomatic foundations of string theory appeared in 1956 – the same year that Church called for such axiomatizations. [ 4 ]
Disjoint-set data structures [9] and partition refinement [10] are two techniques in computer science for efficiently maintaining partitions of a set subject to, respectively, union operations that merge two sets or refinement operations that split one set into two. A disjoint union may mean one of two things. Most simply, it may mean the union ...
In mathematics, a pairing function is a process to uniquely encode two natural numbers into a single natural number. Any pairing function can be used in set theory to prove that integers and rational numbers have the same cardinality as natural numbers. [1]
On the set of real numbers , (,) = + is a binary operation since the sum of two real numbers is a real number. On the set of natural numbers , (,) = + is a binary operation since the sum of two natural numbers is a natural number. This is a different binary operation than the previous one since the sets are different.