urelement or ur-element (from the German prefix ur-, 'primordial') is an object that is not a set (has no elements), but that may be an element of a set....

Look up Ur, ur, ur-, or úr in Wiktionary, the free dictionary. UR, Ur or ur may refer to: A location in Final Fantasy III A race of aliens in Fading Suns...

A {\displaystyle A} is an ur-element. In this case, the usual axiom of extensionality would then imply that every ur-element is equal to the empty set...

In mathematics, an element (or member) of a set is any one of the distinct objects that belong to that set. For example, given a set called A containing...

rule (schema) above, the metavariables A and B can be instantiated to any element of the universe (or sometimes, by convention, a restricted subset such...

Set hereditary Class (Ur-)Element Ordinal number Extensionality Forcing Relation equivalence partition Set operations: intersection union complement Cartesian...

Set hereditary Class (Ur-)Element Ordinal number Extensionality Forcing Relation equivalence partition Set operations: intersection union complement Cartesian...

are sets, then x is an element of the union of M if and only if there is at least one element A of M such that x is an element of A. In symbols: x ∈ ⋃...

} suppose that a is a particular but arbitrarily chosen element of A show that a is an element of B. The validity of this technique can be seen as a consequence...

a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two. More specifically...

product, which can be represented by an n-dimensional array, where each element is an n-tuple. An ordered pair is a 2-tuple or couple. More generally still...

implies x1 = x2). In other words, every element of the function's codomain is the image of at most one element of its domain. The term one-to-one function...

{\displaystyle \lnot \ \exists {x}{\in }\mathbf {X} \,P(x)} . If there is no element of the domain of discourse for which the statement is true, then it must...

can always be considered as a function of a single argument that is an element of some product space. However, it may be convenient for notation to consider...

Set hereditary Class (Ur-)Element Ordinal number Extensionality Forcing Relation equivalence partition Set operations: intersection union complement Cartesian...

guaranteeing that recursively defined functions exist. Given a set X, an element a of X and a function f: X → X, the theorem states that there is a unique...

Set hereditary Class (Ur-)Element Ordinal number Extensionality Forcing Relation equivalence partition Set operations: intersection union complement Cartesian...

Set hereditary Class (Ur-)Element Ordinal number Extensionality Forcing Relation equivalence partition Set operations: intersection union complement Cartesian...

Intuitively, it permits the inference from any conjunction of either element of that conjunction. A {\displaystyle A} and B {\displaystyle B} . Therefore...

unique set with no elements, called the empty set; a set with a single element is a singleton. Sets are uniquely characterized by their elements; this...

any property P: For every element of ∅ {\displaystyle \varnothing } , the property P holds (vacuous truth). There is no element of ∅ {\displaystyle \varnothing...

from the r sequence that does not have a rational difference with any element thus far constructed in the v sequence. Continue until all the reals in...

Set hereditary Class (Ur-)Element Ordinal number Extensionality Forcing Relation equivalence partition Set operations: intersection union complement Cartesian...

Set hereditary Class (Ur-)Element Ordinal number Extensionality Forcing Relation equivalence partition Set operations: intersection union complement Cartesian...

algebra; "true" corresponds to the maximal element of the algebra, and "false" corresponds to the minimal element. Intermediate elements of the algebra correspond...

binary relation f between X and Y is a function (from X to Y) if for every element x in X there is exactly one y in Y such that f relates x to y. The sets...

union, intersection, and complement. Another class of operation is the element-wise application of string operations. Examples: suppose L 1 {\displaystyle...

Set hereditary Class (Ur-)Element Ordinal number Extensionality Forcing Relation equivalence partition Set operations: intersection union complement Cartesian...

Set hereditary Class (Ur-)Element Ordinal number Extensionality Forcing Relation equivalence partition Set operations: intersection union complement Cartesian...

element of the intersection A ∩ B {\displaystyle A\cap B} if and only if x {\displaystyle x} is both an element of A {\displaystyle A} and an element...