Symmetric relation

<table><tbody><tr><th><a href="/facts/Transitive_relation/9r90y50r">Transitive</a> <a href="/facts/Binary_relation/r9BwGgaK">binary relations</a> <ul><li>v</li><li>t</li><li>e</li></ul></th></tr><tr><td><table><tbody><tr><td></td><td> Symmetric</td><td> <a href="/facts/Antisymmetric_relation/YhuqG4z8">Antisymmetric</a></td><td> <a href="/facts/Connected_relation/UpwopAts">Connected</a></td><td> <a href="/facts/Well-founded_relation/zyRRyJ4M">Well-founded</a></td><td> <a href="/facts/Join_and_meet/cYcLhRIR">Has joins</a></td><td> <a href="/facts/Join_and_meet/cYcLhRIR">Has meets</a></td><td> <a href="/facts/Reflexive_relation/NzTytEg9">Reflexive</a></td><td> <a href="/facts/Reflexive_relation/NzTytEg9">Irreflexive</a></td><td> <a href="/facts/Asymmetric_relation/rUdVRwuI">Asymmetric</a></td></tr><tr><td></td><td></td><td></td><td> Total, Semiconnex</td><td></td><td></td><td></td><td></td><td> Anti-reflexive</td><td></td></tr><tr><td> <a href="/facts/Equivalence_relation/1MQ5rtkW">Equivalence relation</a></td><td> Y</td><td> ✗</td><td> ✗</td><td> ✗</td><td> ✗</td><td> ✗</td><td> Y</td><td> ✗</td><td> ✗</td></tr><tr><td> <a href="/facts/Preorder/uubzHSWm">Preorder (Quasiorder)</a></td><td> ✗</td><td> ✗</td><td> ✗</td><td> ✗</td><td> ✗</td><td> ✗</td><td> Y</td><td> ✗</td><td> ✗</td></tr><tr><td> <a href="/facts/Partial_order/qb4a3FAX">Partial order</a></td><td> ✗</td><td> Y</td><td> ✗</td><td> ✗</td><td> ✗</td><td> ✗</td><td> Y</td><td> ✗</td><td> ✗</td></tr><tr><td> <a href="/facts/Total_preorder/LmaM9lSs">Total preorder</a></td><td> ✗</td><td> ✗</td><td> Y</td><td> ✗</td><td> ✗</td><td> ✗</td><td> Y</td><td> ✗</td><td> ✗</td></tr><tr><td> <a href="/facts/Total_order/xnTtmuYJ">Total order</a></td><td> ✗</td><td> Y</td><td> Y</td><td> ✗</td><td> ✗</td><td> ✗</td><td> Y</td><td> ✗</td><td> ✗</td></tr><tr><td> <a href="/facts/Prewellordering/EMhoWkMU">Prewellordering</a></td><td> ✗</td><td> ✗</td><td> Y</td><td> Y</td><td> ✗</td><td> ✗</td><td> Y</td><td> ✗</td><td> ✗</td></tr><tr><td> <a href="/facts/Well-quasi-ordering/suQavGN9">Well-quasi-ordering</a></td><td> ✗</td><td> ✗</td><td> ✗</td><td> Y</td><td> ✗</td><td> ✗</td><td> Y</td><td> ✗</td><td> ✗</td></tr><tr><td> <a href="/facts/Well-order/GrWEBWlW">Well-ordering</a></td><td> ✗</td><td> Y</td><td> Y</td><td> Y</td><td> ✗</td><td> ✗</td><td> Y</td><td> ✗</td><td> ✗</td></tr><tr><td> <a href="/facts/Lattice_(order)/5ak6ufky">Lattice</a></td><td> ✗</td><td> Y</td><td> ✗</td><td> ✗</td><td> Y</td><td> Y</td><td> Y</td><td> ✗</td><td> ✗</td></tr><tr><td> <a href="/facts/Join-semilattice/kW5po0rp">Join-semilattice</a></td><td> ✗</td><td> Y</td><td> ✗</td><td> ✗</td><td> Y</td><td> ✗</td><td> Y</td><td> ✗</td><td> ✗</td></tr><tr><td> <a href="/facts/Meet-semilattice/kW5po0rp">Meet-semilattice</a></td><td> ✗</td><td> Y</td><td> ✗</td><td> ✗</td><td> ✗</td><td> Y</td><td> Y</td><td> ✗</td><td> ✗</td></tr><tr><td> <a href="/facts/Strict_partial_order/qb4a3FAX">Strict partial order</a></td><td> ✗</td><td> Y</td><td> ✗</td><td> ✗</td><td> ✗</td><td> ✗</td><td> ✗</td><td> Y</td><td> Y</td></tr><tr><td> <a href="/facts/Weak_ordering/LmaM9lSs">Strict weak order</a></td><td> ✗</td><td> Y</td><td> ✗</td><td> ✗</td><td> ✗</td><td> ✗</td><td> ✗</td><td> Y</td><td> Y</td></tr><tr><td> <a href="/facts/Strict_total_order/xnTtmuYJ">Strict total order</a></td><td> ✗</td><td> Y</td><td> Y</td><td> ✗</td><td> ✗</td><td> ✗</td><td> ✗</td><td> Y</td><td> Y</td></tr><tr><td></td><td> Symmetric</td><td> <a href="/facts/Antisymmetric_relation/YhuqG4z8">Antisymmetric</a></td><td> <a href="/facts/Connected_relation/UpwopAts">Connected</a></td><td> <a href="/facts/Well-founded_relation/zyRRyJ4M">Well-founded</a></td><td> <a href="/facts/Join_and_meet/cYcLhRIR">Has joins</a></td><td> <a href="/facts/Join_and_meet/cYcLhRIR">Has meets</a></td><td> <a href="/facts/Reflexive_relation/NzTytEg9">Reflexive</a></td><td> <a href="/facts/Reflexive_relation/NzTytEg9">Irreflexive</a></td><td> <a href="/facts/Asymmetric_relation/rUdVRwuI">Asymmetric</a></td></tr><tr><td> Definitions, for all a , b {\displaystyle a,b} and S ≠ ∅ : {\displaystyle S\neq \varnothing :} </td><td> a R b ⇒ b R a {\displaystyle {\begin{aligned}&aRb\\\Rightarrow {}&bRa\end{aligned}}} </td><td> a R b  and  b R a ⇒ a = b {\displaystyle {\begin{aligned}aRb{\text{ and }}&bRa\\\Rightarrow a={}&b\end{aligned}}} </td><td> a ≠ b ⇒ a R b  or  b R a {\displaystyle {\begin{aligned}a\neq {}&b\Rightarrow \\aRb{\text{ or }}&bRa\end{aligned}}} </td><td> min S exists {\displaystyle {\begin{aligned}\min S\\{\text{exists}}\end{aligned}}} </td><td> a ∨ b exists {\displaystyle {\begin{aligned}a\vee b\\{\text{exists}}\end{aligned}}} </td><td> a ∧ b exists {\displaystyle {\begin{aligned}a\wedge b\\{\text{exists}}\end{aligned}}} </td><td> a R a {\displaystyle aRa} </td><td> not  a R a {\displaystyle {\text{not }}aRa} </td><td> a R b ⇒ not  b R a {\displaystyle {\begin{aligned}aRb\Rightarrow \\{\text{not }}bRa\end{aligned}}} </td></tr></tbody></table></td></tr><tr><td>Y indicates that the column's property is always true for the row's term (at the very left), while ✗ indicates that the property is not guaranteed in general (it might, or might not, hold). For example, that every equivalence relation is symmetric, but not necessarily antisymmetric, is indicated by Y in the "Symmetric" column and ✗ in the "Antisymmetric" column, respectively. All definitions tacitly require the <a href="/facts/Homogeneous_relation/wh0QoWzS">homogeneous relation</a> R {\displaystyle R} be <a href="/facts/Transitive_relation/9r90y50r">transitive</a>: for all a , b , c , {\displaystyle a,b,c,} if a R b {\displaystyle aRb} and b R c {\displaystyle bRc} then a R c . {\displaystyle aRc.} A term's definition may require additional properties that are not listed in this table.</td></tr></tbody></table>
A symmetric relation is a type of <a href="/facts/Binary_relation/r9BwGgaK">binary relation</a>. Formally, a binary relation R over a <a href="/facts/Set_(mathematics)/BfucfHMq">set</a> X is symmetric if:

∀
        a
        ,
        b
        ∈
        X
        (
        a
        R
        b
        ⇔
        b
        R
        a
        )
        ,
      
    
    {\displaystyle \forall a,b\in X(aRb\Leftrightarrow bRa),}

where the notation aRb means that (a, b) ∈ R.
An example is the relation "is equal to", because if a = b is true then b = a is also true. If RT represents the <a href="/facts/Converse_relation/EVGlRSz7">converse</a> of R, then R is symmetric if and only if R = RT.
Symmetry, along with <a href="/facts/Reflexive_relation/NzTytEg9">reflexivity</a> and <a href="/facts/Transitive_relation/9r90y50r">transitivity</a>, are the three defining properties of an <a href="/facts/Equivalence_relation/1MQ5rtkW">equivalence relation</a>.

Symmetric relation open-in-new

Symmetric relation