nLab
element

This entry is about the concept in mathematics, specifically in set theory/type theory/category theory. For the concept in chemistry see at chemical element.

Context

Foundations

Mathematics

Contents

Idea

An element of a set is a thing which “belongs to,” or “is an element of,” that set.

The circularity of this definition is unavoidable in foundational set theories in which “set” is an undefined term. In “definitional” set theories, where “set” is defined in terms of something else, elements are likewise defined in terms of the same “something else.”

If sets (or setoids) are regarded as the semantics of some type theory, then an element of a set is the interpretation of a term of some type.

Generalisations

basic symbols used in logic

A\phantom{A}symbolA\phantom{A}A\phantom{A}meaningA\phantom{A}
A\phantom{A}\inA\phantom{A}element relation
A\phantom{A}:\,:A\phantom{A}typing relation
A\phantom{A}==A\phantom{A}equality
A\phantom{A}\vdashA\phantom{A}A\phantom{A}entailment / sequentA\phantom{A}
A\phantom{A}\topA\phantom{A}A\phantom{A}true / topA\phantom{A}
A\phantom{A}\botA\phantom{A}A\phantom{A}false / bottomA\phantom{A}
A\phantom{A}\RightarrowA\phantom{A}implication
A\phantom{A}\LeftrightarrowA\phantom{A}logical equivalence
A\phantom{A}¬\notA\phantom{A}negation
A\phantom{A}\neqA\phantom{A}negation of equality / apartnessA\phantom{A}
A\phantom{A}\notinA\phantom{A}negation of element relation A\phantom{A}
A\phantom{A}¬¬\not \notA\phantom{A}negation of negationA\phantom{A}
A\phantom{A}\existsA\phantom{A}existential quantificationA\phantom{A}
A\phantom{A}\forallA\phantom{A}universal quantificationA\phantom{A}
A\phantom{A}\wedgeA\phantom{A}logical conjunction
A\phantom{A}\veeA\phantom{A}logical disjunction
A\phantom{A}\otimesA\phantom{A}A\phantom{A}multiplicative conjunctionA\phantom{A}
A\phantom{A}\oplusA\phantom{A}A\phantom{A}multiplicative disjunctionA\phantom{A}

Last revised on July 3, 2018 at 02:48:59. See the history of this page for a list of all contributions to it.