Symmetric relation
Type of binary relation / From Wikipedia, the free encyclopedia
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A symmetric relation is a type of binary relation. An example is the relation "is equal to", because if a = b is true then b = a is also true. Formally, a binary relation R over a set X is symmetric if:[1]
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Y indicates that the column's property is always true 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 homogeneous relation be transitive: for all if and then |
where the notation aRb means that (a, b) ∈ R.
If RT represents the converse of R, then R is symmetric if and only if R = RT.[2]
Symmetry, along with reflexivity and transitivity, are the three defining properties of an equivalence relation.[1]