nLab equality

Disambiguation

in logic, higher category theory, and the foundations of mathematics, equality is a notion of when two objects of a collection are considered to be the same objects.

Here is a list of distinctions between different notions of equality, in different contexts, where possibly all the following pairs of notions are, in turn, “the same”, just expressed in terms of different terminologies:

• the difference between axiomatic and defined equality;
• the difference between equality and identity,
• the difference between intensional and extensional equality,
• the difference between propositional equality, judgmental equality, and typal equality, see also equality in type theory
• the difference between equality and isomorphism,
• the difference between equality and equivalence,
• the possibility of operations that might not preserve equality.

Related concepts

• equality, equation

• definition, assignment, equality

• inequality

  • denial inequality

  • apartness relation

  • antithesis interpretation

• isomorphism

• equivalence

• weak equivalence

• homotopy equivalence, weak homotopy equivalence

• equivalence in an (∞,1)-category

• equivalence of (∞,1)-categories

• identification, unique identification

symbol	in propositional logic	Unicode
$:$	typing relation	U+003A
=	propositional equality relation	U+003D
$\neg$	logical negation operator	U+00AC
$\neg \neg$	double negation	U+00AC&U+00AC
$\nLeftarrow$, $\nleftarrow$	negation of converse implication, or negation of converse conditional	U+21CD, U+219A
$\nLeftrightarrow$, $\nleftrightarrow$	negation of logical equivalence, or negation of biconditional	U+21CE, U+21AE
$\nRightarrow$, $\nrightarrow$	negation of implication, or negation of conditional	U+21CF, U+219B
$\Leftarrow$, $\leftarrow$	converse implication, or converse conditional	U+21D0, U+2190
$\Rightarrow$, $\rightarrow$	implication, or conditional	U+21D2, U+2192
$\Leftrightarrow$, $\leftrightarrow$	logical equivalence, or biconditional	U+21D4, U+2192
$\wedge$	logical conjunctionoperator	U+2227
$\vee$	logical dysjunction operator	U+2228
$\neq$	inequality, or apartness relation	U+2260
$\vdash$	syntactic entailment relation	U+22A2
$\vDash$	semantic entailment relation	U+22A8
$\top$	truth value, or top element	U+22A3
$\bot$	false value, or bottom element	U+22A4
$\veebar$, $\oplus$	logical exclusive dysjunction operator	U+22BB, U+2295
$\bar{\wedge}$	logical non-conjunction operator	U+22BC
$\bar{\vee}$	logical non-dysjunction operator	U+22BD

symbol	in first-order logic	Unicode
$\forall$	universal quantifier	U+2200
$\exists$	existential quantifier	U+2203
$\exists!$	uniqueness quantifier	U+2203&U+0021
$\nexists$	negation of existential quantifier	U+2204

symbol	in set theory	Unicode
×	binary Cartesian product, or binary product	U+00D7
$\varnothing$	empty, or uninhabited set	U+2205
$\in$	membership relation	U+2208
$\notin$	negation of membership relation	U+2209
$\ni$	containment relation	U+220B
$\notni$	negation of containment relation	U+220C
$\prod$	n-ary Cartesian product, or product operator	U+220F
$\coprod$	n-ary disjoint union, or coproduct operator	U+2210
$\cap$	binary intersection operator	U+2229
$\cup$	binary union operator	U+222A
$\subset$	subset of relation	U+2282
$\supset$	superset of relation	U+2283
$\nsubset$	negation of subset relation	U+2284
$\nsupset$	negation of superset relation	U+2285
$\subseteq$	inclusion relation, or subset of, or equal to	U+2286
$\supseteq$	converse of inclusion relation, or superset of, or equal to	U+2287
$\sqcup$	binary disjoint union, or binary coproduct operator	U+2294
$\bigcap$	n-ary intersection operator	U+22C2
$\bigcup$	n-ary union operator	U+22C3

$\;$

$\phantom{-}$symbol$\phantom{-}$	$\phantom{-}$in linear logic$\phantom{-}$
$\phantom{A}$$\top$	$\phantom{A}$additive truth
$\phantom{A}$$\bot$	$\phantom{A}$additive falsehood
$\phantom{A}$$0$	$\phantom{A}$multiplicative falsehood
$\phantom{A}$$1$	$\phantom{A}$multiplicative truth
$\phantom{A}$$\multimap$$\phantom{A}$	$\phantom{A}$linear implication$\phantom{A}$
$\phantom{A}$$\otimes$$\phantom{A}$	$\phantom{A}$multiplicative conjunction$\phantom{A}$
$\phantom{A}$$\oplus$$\phantom{A}$	$\phantom{A}$additive disjunction$\phantom{A}$
$\phantom{A}$$\&$$\phantom{A}$	$\phantom{A}$additive conjunction$\phantom{A}$
$\phantom{A}$$\invamp$$\phantom{A}$	$\phantom{A}$multiplicative disjunction$\phantom{A}$
$\phantom{A}$$\;!$$\phantom{A}$	$\phantom{A}$exponential conjunction$\phantom{A}$
$\phantom{A}$$\;?$$\phantom{A}$	$\phantom{A}$exponential disjunction$\phantom{A}$
$\phantom{A}$$^\bot$$\phantom{A}$	$\phantom{A}$negation$\phantom{A}$

$\;$

$\phantom{-}$symbol$\phantom{-}$	$\phantom{-}$in dependent type theory$\phantom{-}$	$\phantom{-}$propositions as types$\phantom{-}$
$\phantom{A}$$\to$	$\phantom{A}$function type	$\phantom{A}$implication
$\phantom{A}$$\times$	$\phantom{A}$product type	$\phantom{A}$conjunction
$\phantom{A}$$+$	$\phantom{A}$sum type	$\phantom{A}$disjunction
$\phantom{A}$$0$, $\emptyset$	$\phantom{A}$empty type	$\phantom{A}$false
$\phantom{A}$$1$	$\phantom{A}$unit type	$\phantom{A}$true
$\phantom{A}$$=$, $\mathrm{Id}$	$\phantom{A}$identity type	$\phantom{A}$propositional equality
$\phantom{A}$$\simeq$	$\phantom{A}$equivalence of types	$\phantom{A}$logical equivalence
$\phantom{A}$$\sum$, $\Sigma$, $\times$	$\phantom{A}$dependent sum type	$\phantom{A}$existential quantifier
$\phantom{A}$$\prod$, $\Pi$, $\to$	$\phantom{A}$dependent product type	$\phantom{A}$universal quantifier
$\phantom{A}$$\mathrm{isContr}$	$\phantom{A}$is contractible type	$\phantom{A}$unique proof
$\phantom{A}$$\mathrm{isContr}(A + B)$	$\phantom{A}$sum type is contractible type	$\phantom{A}$exclusive disjunction
$\phantom{A}$$\mathrm{isContr}\left(\sum_{x:A} B(x)\right)$	$\phantom{A}$dependent sum type is contractible type	$\phantom{A}$uniqueness quantifier
$\phantom{A}$$\mathrm{ua}$	$\phantom{A}$univalence axiom	$\phantom{A}$propositional extensionality

category: disambiguation