The complement(s) of the element 'a' in the lattice shown in the figure (with greatest element I at the top, least element O at the bottom, and elements a, b, c, d, e in between, where O < a < I, O < d < I, O < e < I, and O < c < b < I) is (are) ___________.

GATE 1988 · Discrete Mathematics · Lattice · medium

Answer: The complements of a are b, c, d and e.

1. State the complement condition: the top I and bottom O exist (bounded lattice). We test each non-extreme element x against the two equalities; element a lies in O < a < I and is incomparable to each of b, c, d, e
2. Test b, c, d, e (and reject extremes): since a is incomparable to each of b, c, d, e, the only common upper bound of a and x is I, so a OR x = I, and the only common lower bound is O, so a AND x = O. Thus b, c, d, e are all complements; I and O fail the test, so they are excluded