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The empty set is the set containing no elements. In mathematics, the empty set or void set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. [1] Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other theories, its existence can be deduced.
In mathematics, a relation denotes some kind of relationship between two objects in a set, which may or may not hold. [ 1 ] As an example, " is less than " is a relation on the set of natural numbers; it holds, for instance, between the values 1 and 3 (denoted as 1 < 3), and likewise between 3 and 4 (denoted as 3 < 4), but not between the ...
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. A simpler example is equality. Any number is equal to itself (reflexive). If , then (symmetric).
Reflexive relation. In mathematics, a binary relation on a set is reflexive if it relates every element of to itself. [1][2] An example of a reflexive relation is the relation "is equal to" on the set of real numbers, since every real number is equal to itself. A reflexive relation is said to have the reflexive property or is said to possess ...
In axiomatic set theory, the axiom of empty set, [1][2] also called the axiom of null set[3] and the axiom of existence, [4][5] is a statement that asserts the existence of a set with no elements. [3] It is an axiom of Kripke–Platek set theory and the variant of general set theory that Burgess (2005) calls "ST," and a demonstrable truth in ...
Partial orders. A reflexive, weak, [1] or non-strict partial order, [2] commonly referred to simply as a partial order, is a homogeneous relation ≤ on a set that is reflexive, antisymmetric, and transitive. That is, for all it must satisfy: Reflexivity: , i.e. every element is related to itself.
Three sets involved. [edit] In the left hand sides of the following identities, L{\displaystyle L}is the L eft most set, M{\displaystyle M}is the M iddle set, and R{\displaystyle R}is the R ight most set. Precedence rules. There is no universal agreement on the order of precedenceof the basic set operators.
Statement. A choice function (also called selector or selection) is a function f, defined on a collection X of nonempty sets, such that for every set A in X, f (A) is an element of A. With this concept, the axiom can be stated: Axiom — For any set X of nonempty sets, there exists a choice function f that is defined on X and maps each set of X ...