Hasse Diagram

Partially Ordered Set (POSET): 

                                   A relation which is reflexive, anti symmetric and transitive is called POSET.

Example: >= and <= are examples of POSET

  Example:
             let A={1,2,3} and the relation R is defined on A such that R={(1,1),(2,2),(3,3),(1,2),(2,3),(1,3)} is a partially ordered set.

R is reflexive since (1,1),(2,2),(3,3) belongs to R\

R is ant symmetric for example (1,2) belongs to R and (2,1) does not belong to R. Similarly (2,3)
belongs to R and (3,2) does not belong to R .

R is transitive since (1,2) and (2,3) belongs to R then (2,3) belongs to R

So R is Partially Ordered Set

Hasse Diagram:
                          Hasse diagram is used to represent POSET in a diagramatic way.

For two elements a and b, b is said to be immediate successor of a if and only if  there is no intermediate element between a and b.

Two elements are said to be comparable if the two elements can be compared by using less than or greater than.

If two elements are on the same level and the two elements are incomparable then the two elements are said to be incomparable

Draw the Hasse Diagram for the divisibility on the set { 1,2,3,6,8,12}

The following are the ordered pairs in divisibility set {(1,1),(2,2),(3,3),(6,6),(8,8),(12,12),(1,2),(1,3),(1,6),(1,8),(1,12),(2,6),(2,8),(2,12),(3,6),(3,12),(6,12)}

Since 1 divides 3 , 1 and 3 are on the same line and 3 divides 6, 3 and 6 are on the same line. Since the given relation is transitive 1 divides 3 and 3 divides 6 so 1 divides 6. Similarly 1 divides 12 and so on.

Lower Bound of above Hasse diagram is 1 and Upper Bound of above Hasse Diagram is 12



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