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A set of polygons in an Euler diagram This set equals the one depicted above since both have the very same elements.. In mathematics, a set is a collection of different [1] things; [2] [3] [4] these things are called elements or members of the set and are typically mathematical objects of any kind: numbers, symbols, points in space, lines, other geometrical shapes, variables, or even other ...
Set-builder notation can be used to describe a set that is defined by a predicate, that is, a logical formula that evaluates to true for an element of the set, and false otherwise. [2] In this form, set-builder notation has three parts: a variable, a colon or vertical bar separator, and a predicate. Thus there is a variable on the left of the ...
The term is commonly used in mathematics and computer science to refer to a listing of all of the elements of a set. The precise requirements for an enumeration (for example, whether the set must be finite, or whether the list is allowed to contain repetitions) depend on the discipline of study and the context of a given problem.
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.
Euler diagram illustrating that the set of "animals with four legs" is a subset of "animals", but the set of "minerals" is a disjoint set (it has no members in common) with "animals" Euler diagram showing the relationships between different Solar System objects
A set is described by listing elements separated by commas, or by a characterizing property of its elements, within braces { }. [8] Since sets are objects, the membership relation can relate sets as well, i.e., sets themselves can be members of other sets. A derived binary relation between two sets is the subset relation, also called set inclusion.
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 ...
In Quine's set-theoretical writing, the phrase "ultimate class" is often used instead of the phrase "proper class" emphasising that in the systems he considers, certain classes cannot be members, and are thus the final term in any membership chain to which they belong. Outside set theory, the word "class" is sometimes used synonymously with "set".
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