To draw the Hasse diagram for the given set of ordered pairs, let's first analyze the given relations. We are provided with the following set of relations:

$\{(a, a), (a, b), (a, c), (a, d), (b, b), (b, c), (b, d), (c, c), (d, d)\}$

Step-by-Step Process to Create the Hasse Diagram:

1. Understand the Partial Order:

   • The relation appears to be a partial order on the elements $\{a, b, c, d\}$, where each pair indicates $x \leq y$.
   • The elements are ordered such that $a \leq b \leq c \leq d$, because we see:
     • $a \leq b$, $a \leq c$, $a \leq d$
     • $b \leq c$, $b \leq d$
     • $c \leq d$

2. Remove Reflexive Pairs:

   • Reflexive pairs, like $(a, a)$, $(b, b)$, $(c, c)$, $(d, d)$, are not drawn in the Hasse diagram, since they are implied for every element in the partial order.

3. Remove Transitive Pairs:

   • In a Hasse diagram, transitive relations are not drawn. For example:
     • $a \leq c$ can be inferred from $a \leq b$ and $b \leq c$, so we remove $(a, c)$.
     • Similarly, $a \leq d$ can be inferred via $a \leq b$ and $b \leq d$, so we remove $(a, d)$.
     • Also, $b \leq d$ can be inferred from $b \leq c$ and $c \leq d$, so we remove $(b, d)$.

4. Remaining Pairs for the Hasse Diagram:

   • After removing reflexive and transitive pairs, the remaining pairs are: $\{(a, b), (b, c), (c, d)\}$

Hasse Diagram:

The Hasse diagram is a simple chain where $a$ is at the bottom and $d$ is at the top, and the elements are connected in a linear fashion:

d
|
c
|
b
|
a

This diagram shows that:

• $a \leq b$,
• $b \leq c$,
• $c \leq d$.

Would you like further clarification or details on how the diagram was constructed?

Related Questions:

1. What are reflexive, antisymmetric, and transitive properties in partial orders?
2. How does a Hasse diagram represent partial orders more clearly than a general diagram?
3. What is the difference between a total order and a partial order?
4. Can a Hasse diagram have cycles, or must it always be acyclic?
5. How can the Hasse diagram be used to find the maximal and minimal elements of a set?

Tip:

Always remove reflexive and transitive pairs when constructing a Hasse diagram to simplify the representation.