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A "✓" indicates that the column property is required in the row definition. For example, the definition of an equivalence relation requires it to be symmetric. All definitions tacitly require transitivity and reflexivity. |
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:
If RT represents the converse of R, then R is symmetric if and only if R = RT.
Symmetry, along with reflexivity and transitivity, are the three defining properties of an equivalence relation.
By definition, a nonempty relation cannot be both symmetric and asymmetric (where if a is related to b, then b cannot be related to a (in the same way)). However, a relation can be neither symmetric nor asymmetric, which is the case for "is less than or equal to" and "preys on").
Symmetric and antisymmetric (where the only way a can be related to b and b be related to a is if a = b) are actually independent of each other, as these examples show.
Symmetric | Not symmetric | |
Antisymmetric | equality | "is less than or equal to" |
Not antisymmetric | congruence in modular arithmetic | "is divisible by", over the set of integers |
Symmetric | Not symmetric | |
Antisymmetric | "is the same person as, and is married" | "is the plural of" |
Not antisymmetric | "is a full biological sibling of" | "preys on" |