Hasse Diagram is created for POSET or Partially Ordered Set. It means that there is a set of elements in which certain element are ordered, sequenced or arranged in some way. It is usually denoted as** ≤, **this is not “Less than, Equal to”, this symbol shows that elements are ordered.

Now, there is a relation among all the elements in the partial order set and it is some binary relation R, this relation must satisfy following properties

- Reflexive
- Anti-Symmetric
- Transitive

### Example Problem

#### Reflexive Property

#### Anti-Symmetric Property

#### Transitive Property

### Hasse Diagram

#### Rules for Hasse Diagram

- If x < y, then in the graph x appears lower to y.
- We draw line segment between x and y only if x cover y or y cover x, it means some order is maintained between them.

#### Step 1

Due to the reflexive property all elements have direct edge to itself.

In this diagram, it shows the relations removed all the self-directing loops.

#### Step 2

The green lines shows transitivity, so we remove all the transitive lines. Out diagram look like the following.

#### Step 3

Replace all the vertices with dots and directed edges with ordinary lines. This final diagram is called the Hasse Diagram of poset.

### Exercise

Here is an exercise for you to practice. The prerequisite for Hasse Diagram is to know how to represent relations using graphs.

Let A be a poset, A = { 2, 4, 6, 8 } and the relation a | b is ‘a divides b. Draw a Hasse Diagram for the poset showing all the relations.