Must an asymmetric relation also be antisymmetric? Must an antisymmetric relation be asymmetric? Give reasons for your answers.
For a relation on a set
- asymmetric if implies that ,
- antisymmetric if whenever .
Every asymmetric relation is antisymmetric.
Proof: Suppose is an asymmetric relation on a set . Let . Then, and . Hence, is antisymmetric.
An antisymmetric relationship need not be asymmetric. For example,
is anti symmetric but not asymmetric on the set
is anti - symmetric as well as asymmetric on the set .
Use quantifiers to express what it means for a relation to be asymmetric.