Question

8. We consider a set B, as B = {1,5,7,9} and perform the Cartesian product of set B with itself. From this Cartesian product, let us create a partially ordered set R having the pairs of elements, as R = {(i, j) | i<=j}. Represent set R as a graph in the form of (a) directed graph and (b) Hasse diagram. (12pts)

Answer 1

The answer for the above mentioned question is explained below :

Given set is B = { 1,5,7,9 }

Cartesian product of set B with itself is as follows :

B * B = { (1,1) , (1,5) , (1,7) , (1,9) , (5,1) , (5,5) , (5,7) , (5,9) , (7,1) , (7,5) , (7,7) , (7,9) , (9,1) , (9,5) , (9,7) , (9,9) }

Given Relation R is as follows ::

R = {(i,j) | i<=j}

Therefore the partial ordered set by using the above relation R is as follows ::

R = { (1,1),(5,5),(7,7),(9,9),(1,5),(1,7),(1,9),(5,7),(5,9),(7,9) }

a) Directed graph for R is as follows ::

In directed graph representation every directed edge from i to j represent (i ,j) in R .

b) Hasse diagram for R is as follows ::

In Hasse diagram representation , the elements form down to up are related i.e., they are in the set R and we hide here Reflexive and Transitive representation of the set R because we know that in every Partially ordered set it must be Reflexive and Transitive so we hide here to make the Hasse diagram simple and understanding .

4, 11

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