# Symmetric Relation

## Symmetric Relation

Relation defined in a set E so that, for every ordered pair of elements (x, y) of $$E\times E$$, if x is in relation with y, then y is in relation with x.

• The arrow diagram of a symmetric relation in a set E includes a return arrow every time that there is an arrow going between two elements.
• A relation defined in a set E so that, for every ordered pair (x, y) of E $$\times$$ E, with x ≠ y, (yx) is not an ordered pair of the relation, is called an antisymmetric relation.
• A relation defined in a set E that is neither symmetric nor antisymmetric is a non-symmetric relation.
• A relation defined in a set E so that, for all pairs of elements {x, y}, either one of the ordered pairs (x, y) or (y, x) belong to the relation, but never both at the same time, is an asymmetric relation.

### Examples

• In a set of lines of a plane, the relation “…is perpendicular to…” is a symmetric relation.
• In a set of numbers, the relation “…divides…” is an antisymmetric relation.
• In a set of numbers, the relation “…is less than…” is an asymmetric relation.