GATE CS 1993 Q8.5 — Partial Orders and Lattices

To determine if the strict less-than relation ($<$) on real numbers ($\mathbb{R}$) is a partial ordering, we evaluate the standard properties of a partial order relation (which must be reflexive, antisymmetric, and transitive):

1. Reflexivity: A relation $R$ is reflexive if $a R a$ for all $a \in \mathbb{R}$. For the relation $<$, $a < a$ is false for any real number $a$. Thus, $<$ is not reflexive (it is irreflexive).
2. Antisymmetry: A relation $R$ is antisymmetric if $a R b$ and $b R a \implies a = b$. For $<$, the premise $a < b$ and $b < a$ can never be simultaneously true. Therefore, the implication holds vacuously. Thus, $<$ is antisymmetric.
3. Asymmetry: A relation $R$ is asymmetric if $a R b \implies b \not R a$. Since $a < b \implies b \not< a$ is always true, $<$ is asymmetric.

Since the relation $<$ is not reflexive, it is not a partial ordering.

Now let's evaluate the given options:

• A & B: Incorrect, because $<$ is not a partial ordering and is not reflexive.
• C: Incorrect, because it claims $<$ is "not asymmetric", whereas $<$ is indeed asymmetric.
• D: Incorrect, because it claims $<$ is "not antisymmetric" and "reflexive", whereas $<$ is antisymmetric and not reflexive.

Since none of the statements (A, B, C, or D) are correct, the correct choice is E.