Relation $<$ is :
- Irreflexive and hence not reflexive
- Asymmetric and hence antisymmetric and also not symmetric
The relation is not POSET because it is irreflexive.
Condition for Antisymmetric: $\forall a,b \in \mathbb{R}, aRb \neq bRa$ unless $a=b.$
For asymmetric we have a stronger requirement excluding the unless part from the antisymmetric requirement. i.e., $\forall a,b \in \mathbb{R}, aRb \neq bRa$
A relation may be 'not Asymmetric and not reflexive' but still Antisymmetric. Example: $\{(1,1),(1,2)\}$ over the set $\{1,2\}$ which is
- not reflexive because $(2,2)$ is not present
- not irreflexive because $(1,1)$ is present
- not symmetric because $(1,2)$ is present and $(2,1)$ is not
- not asymmetric because $(1,1)$ is present
- but is anti-symmetric.
Antisymmetric and Irreflexive $=$ Asymmetric
Correct Option E.