Q1. Let {1,2,3,4,6,9} with the partial order of divisibility. Draw its Hasse Diagram Q2. [{1,2,3,4,5}, s], Draw its Hasse Diagram

Question 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 to a relation

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 preceded 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 one element that succeeds all other elements.

Question 2

$S= \{1,2,3,4,5\}\\ R=\{(4,2),(3,4),(5,1)\}$

Smallest partial order relation

$\{(1,5),(2,3,4)\}$

Hasse diagram for poset