Relations: Partial Ordering and Hasse Diagram

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Joshua Capel was given access to the Relations: Partial Ordering and Hasse Diagram 7 years, 8 months ago

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Joshua Capel 7 years, 8 months ago

Created this as a copy of Testing JSXGraph for a Hasse diagram.

Name	Status	Author	Last Modified
Testing JSXGraph for a network diagram	Needs to be tested	Joshua Capel	07/08/2017 01:51
Testing JSXGraph for a single edge	draft	Joshua Capel	03/08/2017 07:15
Testing JSXGraph for a Hasse diagram	Needs to be tested	Joshua Capel	13/08/2017 01:30
Relations: Partial Ordering and Hasse Diagram	draft	Daniel Mansfield	06/06/2021 17:15

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Question statement

Another important relation is called a partial order. We say that a relation $\leq$ is a partial order when it has the three characteristic qualities:

$a\leq a$ always
if $a\leq b$ and $b \leq a$ then $b=a$
if $a \leq b$ and $b \leq c$ then $a \leq c$ .

Typically, we take a set together with partial ordering such as "less than or equal to" $\leq$ , "subset" $\subseteq$ or "divides" $|$ . Together, the partial ordering and set are called a partially ordered set or poset.

Instead of an arrow diagram, we use a Hasse Diagram to visualise a poset. This communicates the relation between elements of the set with as few edges as possible.

Consider the set

$A = \left\{ \left\{1\right\} ,\left\{2 \right\} ,\left\{3 \right\},\left\{1,2 \right\} ,\left\{ 3,4\right\},\left\{1,2,4 \right\},\left\{1,3,4 \right\},\left\{1,2,4,5 \right\},\left\{ 1,2,3,5 \right\} \right\}$

together with the partial ordering $\subseteq$ .

Give any introductory information the student needs.

			Name	Type	Generated Value

			a	number	0.9
			b	number	0.3

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Generated value: `number`

0.9

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Ask the student a question, and give any hints about how they should answer this part.

Make a Hasse Diagram for this poset. Click on two verticies to make or destroy the edge between them.

{createNetwork()}

Advice

See the steps section above for a general description for how to design Hasse diagrams.

A good rule of thumb is to work from the bottom up. Join all 1-element sets to those 2-element sets that contain them as a subset. Then do the same for 2-element sets to 3-element sets, and 3-element sets to 4-element sets.

Next check if any of the 1-element sets are contained in any of the 3-element sets that they are not already connected to by a path of edges. If they are, join these vertices. Do the same for 2-element sets contained in 4-element sets.

Finally, check if any of the 1-element sets are contained in any of the 4-element sets that they are not already connected to by a path of edges.

Give a worked solution to the whole question.

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This question is used in the following exam:

Week 4: Hasse diagrams and logical statements by Daniel Mansfield in Discrete Mathematics.

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Relations: Partial Ordering and Hasse Diagram

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Joshua Capel was given access to the Relations: Partial Ordering and Hasse Diagram 7 years, 8 months ago

Joshua Capel 7 years, 8 months ago

Error in variable testing condition

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Generated value: `number`

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Relations: Partial Ordering and Hasse Diagram

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Joshua Capel was given access to the Relations: Partial Ordering and Hasse Diagram 7 years, 8 months ago

Joshua Capel 7 years, 8 months ago

Error in variable testing condition

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Warning

Generated value: number

Parts

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Generated value: `number`