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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 ...
Let R be the set of all sets that are not members of themselves. (This set is sometimes called "the Russell set".) If R is not a member of itself, then its definition entails that it is a member of itself; yet, if it is a member of itself, then it is not a member of itself, since it is the set of all sets that are not members of themselves. The ...
To investigate the left distributivity of set subtraction over unions or intersections, consider how the sets involved in (both of) De Morgan's laws are all related: () = = () always holds (the equalities on the left and right are De Morgan's laws) but equality is not guaranteed in general (that is, the containment might be strict).
In abstract algebra, the free monoid on a set is the monoid whose elements are all the finite sequences (or strings) of zero or more elements from that set, with string concatenation as the monoid operation and with the unique sequence of zero elements, often called the empty string and denoted by ε or λ, as the identity element.
For example, the union of three sets A, B, and C contains all elements of A, all elements of B, and all elements of C, and nothing else. Thus, x is an element of A ∪ B ∪ C if and only if x is in at least one of A, B, and C. A finite union is the union of a finite number of sets; the phrase does not imply that the union set is a finite set ...
Set is the prototype of a concrete category; other categories are concrete if they are "built on" Set in some well-defined way. Every two-element set serves as a subobject classifier in Set. The power object of a set A is given by its power set, and the exponential object of the sets A and B is given by the set of all functions from A to B.
More generally, a collection of any sets whatsoever is called a family of sets, set family, or a set system. Additionally, a family of sets may be defined as a function from a set I {\displaystyle I} , known as the index set, to F {\displaystyle F} , in which case the sets of the family are indexed by members of I {\displaystyle I} . [ 1 ]
The expressions "A includes x" and "A contains x" are also used to mean set membership, although some authors use them to mean instead "x is a subset of A". [2] Logician George Boolos strongly urged that "contains" be used for membership only, and "includes" for the subset relation only.