Answer to Question #117156 in Discrete Mathematics for Priya

Question

Draw the Hasse diagram for divisibility on the set {1,2,3,4,6,8,12}. Do the maximal, minimal elements exist? If so, what are they? What is the greatest element?

Answer 1

Hasse diagram is a graphical representation of a partially ordered set. A={1, 2, 3, 4, 6, 8, 12}, the relation R="divisibility".

R={(1,1), (1,2), (1,3), (1,4), (1,6), (1,8), (1,12), (2,2), (2,4), (2,6), (2,8), (2,12), (3,3), (3,6), (3,12), (4,4), (4,8), (4,12), (6,6), (6,12), (8,8), (12,12)}.

Step 1. We construct a directed graph corresponding a relation R.

Step 2. We remove all loops from the diagram (reflexivity) and all transitive edges.

Step 3. We make sure that the initial vertex is below the terminal vertex and remowe all arrows. See Hasse diagram:

The minimal element is 1 (not preceeded by another element).

The maximal elements are 8 and 12 (not succeeded by another element).

The greatest element does not exist since there is no any one element that succeeds all other elements.