Are these definitions correct?
Irreflexive:
A relation is irreflexive If For some a, a is not related to a
Anti-reflexive:
For all a, a Is not related to a.
Are the definitions for asymmetric and anti-symmetric similar?
Asked
Active
Viewed 515 times
2 Answers
1
No.
The terms Irreflexive and Antireflexive are synonyms.
A relation is called such when it does not relate any element to itself.
$\qquad\forall x\in A~.\langle x,x\rangle\notin R$
The definitions for asymmetric and anti-symmetric are distinct.
Asymmetric relations are those where a pair and its inverse are never both in the relation.
$\qquad\forall x\in A~\forall y\in A~.\big(\langle x,y\rangle\in R\to\langle y,x\rangle\notin R\big)$
Antisymmetric relations are those where if a pair does not have identical members then it and its inverse are never both in the relation.
$\qquad\forall x\in A~\forall y\in A~.\big(\langle x,y\rangle\in R\land (x\neq y)\to \langle y,x\rangle\notin R\big)$
Which is more usually expressed as: when any pair and its inverse are in the relation, that pair has identical members.
$\qquad\forall x\in A~\forall y\in A~.\big(\langle x,y\rangle\in R\land\langle y,x\rangle\in R\to (x=y)\big)$
A relation is Asymmetric when it is both Antisymetric and Irreflexive.
Graham Kemp
- 129,094
-
Thanks. I guess I was thinking that some of these terms simply indicated a violation of reflexivity or symmetry for some pair. But in those cases we should simply say that the relation is not reflexive or not symmetric which does not make the stronger claim that property does not hold for any pair, as these terms are intended to do. – Darrel Lee Jul 21 '22 at 00:27
-
This question is nearly identical: https://math.stackexchange.com/q/778164/340418 – Darrel Lee Jul 21 '22 at 00:38
0
These terms are synonyms for another: “Anti” is also used to describe the relation which is a “strong” relation opposite:
Ref: [∀a: (a,a)∈R] AntiRef: [∀a: (a,a)∉R]
Sym: [∀a,b: (a,b)∈R ∧ (b,a)∈R] AntiSym: [∀a,b: (a,b)∈R ∧ (b,a)∉R] however, what if a=b? [(a,a)∈R ∧ (a,a)∉R] is a contradiction— this special case is the reason for “AntiSymmetry” and “Asymmetry”.
Asymmetry does not allow for a=b, it is “Irreflexive” and “AntiSymmetric”— whereas, AntiSym does allow for a=b.
This alternative naming (which is confusing), highlights the logical interpretation of relations (properties change the name): “Irreflexive” probably gets its name from another logical property, which is why it isn’t just “AntiReflexive”.
One confusing example: “Intransitive” is very different from “AntiTransitive”: it is not a “strong” anti-relation. “Intransitivity” is the negation of the entire statement (NOT “for all” triples -> “there exists” one triple…).
In summary: “Irreflexive” is synonymous with “Antireflexive” (probably because irreflexive has another special property).
Cheers
