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Q22E

Expert-verifiedFound in: Page 631

Book edition
7th

Author(s)
Kenneth H. Rosen

Pages
808 pages

ISBN
9780073383095

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

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

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

** 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).**

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.

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