Basic concepts in set theory

association of a single output to each input

set (in mathematics) containing no elements

150px|thumb|right|class=skin-invert-image|Euler diagram showing A is a [[subset of B (denoted A \subseteq B) and, conversely, B is a superset of A (denoted B \supseteq A)]]

operation denoted by symbol “∪” applied on two sets; the set of all distinct elements in the collection

function that "reverses" another function: if the function f applied to an input x gives a result of y, then applying its inverse function g to y gives the result x, and vice versa. i.e., f(x) = y if and only if g(y) = x

concept in set theory (for the term in geometry, see Q1364910)

In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equivalently, a bijection is a relation between two sets such that each element of either set is paired with exactly one element of the other set.

mathematical function that preserves distinctness

function such that every element of the codomain has a preimage

set of "input" or argument values for which a function is defined

set that has a finite number of elements

operation which takes two mathematical functions and makes one function of these

pair of mathematical objects; tuple of specific length (tuple length n=2)

unary operation on sets: the set of non-elements of the argument set

any one of the distinct objects that make up a set in set theory

function that always returns the same value that was used as its argument

sets with no element in common

set of all values of a function; set of all values that a function can produce; subset of codomain

In mathematics, a tuple is a finite sequence (or ordered list) of numbers. More generally, it is a sequence of mathematical objects, called the elements of the tuple. An -tuple is a tuple of elements, where is a non-negative integer. There is only one 0-tuple, called the empty tuple. A 1-tuple and a 2-tuple are commonly called a singleton and an ordered pair, respectively. The term "infinite tuple" is occasionally used for "infinite sequences".

mathematical ways to group elements of a set

right|thumb|250px|A function from to . The blue oval is the codomain of . The yellow oval inside is the Image (mathematics)|image of , and the red oval is the domain of .

function that returns 1 if an element is present in a specified subset and 0 if absent; naturally isomorphic with a set's subsets

mathematical definition in set theory

set with exactly one element

concept of mathematical logic

In mathematics, a multiset (or bag, or mset) is a modification of the concept of a set that, unlike a set, allows for multiple instances for each of its elements. The number of instances given for each element is called the multiplicity of that element in the multiset. As a consequence, an infinite number of multisets exist that contain only elements and , but vary in the multiplicities of their elements: The set contains only elements and , each having multiplicity 1 when is seen as a multiset. In the multiset , the element has multiplicity 2, and has multiplicity 1. In the multiset , and

function, sometimes assumed structure-preserving in a proper sense

collection of objects, each associated with an index from some index set

mathematical identities and relationships involving sets

modified union operation that indexes the elements according to which set they originated in

ambiguous term, referring either to the codomain or the image of a function

collection of some of the subsets of a set; collection of any sets whatsoever

in set theory, a set which contains all objects, including itself

category in mathematics where the objects are sets and the morphisms are the total functions between the sets

mathematical term

or inclusion function, or canonical injection

set having two elements a and b with no particular relation between them

inverse image of a singleton in the field of mathematics

test for the bijectivity of a function

a set equipped with a choice of a specific element

mathematical function f that is defined on some collection X of nonempty sets and assigns some element of each set S in that collection to S by f(S)

an operation in set theory that maps elements in a set to elements from another set

Injection: one to one, surjection: onto, bijection: both one to one and onto.

function whose domain is a collection of sets