GATE CS 1997 Q3.3 — Partial Orders and Lattices

In any lattice with a greatest element $I$ and a least element $O$, an element $y$ is a complement of $x$ if and only if:

From the given Hasse diagram:

• The least element (bottom) of the lattice is $a$.
• The greatest element (top) of the lattice is $g$.

To find the complements of $e$, we look for elements $x$ such that $e \wedge x = a$ and $e \vee x = g$:

1. For $x = b$: $e \wedge b = a$ and $e \vee b = g$
2. For $x = c$: $e \wedge c = a$ and $e \vee c = g$
3. For $x = d$: $e \wedge d = a$ and $e \vee d = g$

Thus, the element $e$ has exactly three complements: $b$, $c$, and $d$.