GATE CS 1999 Q2.3 — Sets, Relations and Functions

Let us analyze the given properties of the relation $R$ on the set $L$:

1. Reflexive, Anti-symmetric, and Transitive: By definition, a relation $R$ on a set $L$ that is reflexive, anti-symmetric, and transitive is a partial order relation. Therefore, the pair $(L, R)$ is a partially ordered set (poset).
   Hence, Option A is true.


2. Existence of LUB and GLB: A poset $(L, R)$ is defined as a lattice if every pair of elements $a, b \in L$ has a unique least upper bound ($lub(a, b)$) and a unique greatest lower bound ($glb(a, b)$). Since the problem states that $lub(a, b)$ and $glb(a, b)$ exist for any two elements in $L$, $L$ is indeed a lattice.
   Hence, Option C is true.


3. Boolean Algebra: A lattice is a Boolean algebra if and only if it is both complemented and distributive. The given conditions do not guarantee distributivity or the existence of complements (for example, any non-distributive lattice like $M_3$ or $N_5$ satisfies the given conditions but is not a Boolean algebra).
   Hence, Option B is not necessarily true.

Thus, the correct options are A and C.