Reflexive does not imply symmetric.
The following relation is reflexive, but not symmetric:
The union of two reflexive relations is reflexive.
The union of two symmetric relations is symmetric.
The union of two transitive relations is not always transitive.
7917Union of Transitive Relations Not Always Transitive
The inverse of an inverse relation is the relation itself.
7920Inverse of Inverse Relation
A relation which is symmetric, transitive, and serial is also reflexive.