Question

To draw the Hasse diagram for divisibility on the set \(\{ 1,3,9,27,81,243\} \).

Short answer

The Hasse diagram for divisibility on the set \(\{ 1,3,9,27,81,243\} \) is drawn as

Step-by-step solution

Given data

For divisibility on the set \(\{ 2,3,5,10,11,15,25\} \)

Concept used Hasse diagram

Hasse diagrams are obtained from a directed graph of a partial ordering by,

1) Removing all loops due to reflexivity from the graph of a partial ordering.

2) Removing all edges occurring due to transitivity of the partial ordering.

3) Arranging all edges to point upwards and deleting (directional) arrows

Thus to get all ordered pairs ordered pairs in the partial ordering for a given Hasse diagram, we look for pairs\((x,y)\)such that path from\(x\)to\(y\)is going upwards . In addition, we also need to add pairs of the form\((x,x)\)to account for reflexive pairs (loops).

Draw the Hasse diagram

Consider for divisibility on the set \(\{ 1,3,9,27,81,243\} \).

The Hasse diagram for for divisibility on the set \(\{ 1,3,9,27,81,243\} \)

From the above Hasse diagram it is clear that, the every number divides next number in the set. Hence, The Hasse diagram for divisibility on the set \(\{ 1,3,9,27,81,243\} \) is drawn.