Question

In Exercises 25–27 list all ordered pairs in the partial ordering with the accompanying Hasse diagram.

25.

Short answer

The list of all odered pairs are \(\{ (a,a),(b,b),(c,c),(d,d),(a,b),(a,c),(a,d),(b,c),(b,d)\} \).

Step-by-step solution

Given data

The Hasse diagram

Concept used Hasse diagrams

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).

Find the ordered pairs

The above hasse diagram is on the set \(P = \{ a,b,c,d\} \)

The relation is given by \(R = \{ (a,a),(b,b),(c,c),(d,d),(a,b),(a,c),(a,d),(b,c),(b,d)\} \)

Hence, The list of all ordered pairs are \(R = \{ (a,a),(b,b),(c,c),(d,d),(a,b),(a,c),(a,d),(b,c),(b,d)\} \).