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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.
A class that is not a set (informally in Zermelo–Fraenkel) is called a proper class, and a class that is a set is sometimes called a small class. For instance, the class of all ordinal numbers , and the class of all sets, are proper classes in many formal systems.
Set theory is the branch of mathematics that studies sets, which are collections of objects, such as {blue, white, red} or the (infinite) set of all prime numbers. Partially ordered sets and sets with other relations have applications in several areas. In discrete mathematics, countable sets (including finite sets) are the main focus
Intersections of the unaccented modern Greek, Latin, and Cyrillic scripts, considering only the shapes of the letters and ignoring their pronunciation Example of an intersection with sets. The intersection of two sets and , denoted by , [3] is the set of all objects that are members of both the sets and .
In mathematics, the symmetric difference of two sets, also known as the disjunctive union and set sum, is the set of elements which are in either of the sets, but not in their intersection. For example, the symmetric difference of the sets { 1 , 2 , 3 } {\displaystyle \{1,2,3\}} and { 3 , 4 } {\displaystyle \{3,4\}} is { 1 , 2 , 4 ...
The syntax of the Python programming language is the set of rules that defines how a Python program will be written and interpreted (by both the runtime system and by human readers). The Python language has many similarities to Perl, C, and Java. However, there are some definite differences between the languages.
Also, there are two membership relations: the first, denoted by "∈", is between two sets; the second, denoted by "η", is between a set and a class. [2] This redundancy is required by many-sorted logic because variables of different sorts range over disjoint subdomains of the domain of discourse. The differences between these two approaches ...
For example, to model the general concept of "equality" as a binary relation =, take the domain and codomain to be the "class of all sets", which is not a set in the usual set theory. In most mathematical contexts, references to the relations of equality, membership and subset are harmless because they can be understood implicitly to be ...