In a lattice defined by the Hasse diagram given in figure, how many…

To find the complements of e quickly, we look for any element x that satisfies two conditions with e:

1. Meet (Bottom): They must only intersect at the very bottom (f).

2. Join (Top): Their paths upward must meet for the first time at the very top (a).

Here is the direct analysis for element e:

• Elements b, a, and f fail immediately because b and a are directly above e (violating the bottom condition), and f is directly below e (violating the top condition).

This leaves us to check the entire right and middle branches: g, c, and d.

• Checking g: * Going down: e and g only meet at f.

  • Going up: The paths from e (via b) and g meet for the first time at a.

  • Result: g is a complement.

• Checking c: * Going down: e and c only meet at f.

  • Going up: The paths from e (via b) and c meet for the first time at a.

  • Result: c is a complement.

• Checking d: * Going down: e and d only meet at f.

  • Going up: The paths from e (via b) and d (via c) meet for the first time at a.

  • Result: d is a complement.

Since g, c, and d all satisfy both conditions, e has exactly 3 complements.