Partial Orders


















Example: prerequisite

Consider the domain of computer science courses.

Suppose a relation is defined on this domain such that
the pair (x,y) is in the relation if x is a prerequisite for y.
(Call this the 'prerequisite' relation.)

Some of the pairs in the relation are:

(111,235) (235,236) (236,240) (240,312)
(240,330) (240,340) (330,431) (340,431)

Is the 'prerequisite' relation reflexive? antisymmetric? transitive?


















What's a Partial Order?

	A relation that's reflexive, antisymmetric, and transitive.

What key idea is represented by partial orders?

	order


















Draw the graph representation of the relation on the board.

	T = {(1,1),(2,2),(3,3),(4,4),(1,2),(1,3),(2,4),(3,4),(1,4)}

If you know the relation is a partial order, can you draw the graph
with fewer edges and not lose any information?

What's a Hasse diagram?

	1. graph representation of a partial order
	2. without reflexive and transitive edges


















Classwork
You may work with a partner.

For each relation (defined on {1,2,3,4}):

1. State whether the relation is a partial order or not.
2. If it is a partial order, give a Hasse diagram.
3. If not, list the properties that are lacking.

	P = {(1,1),(2,2),(3,3),(4,4)}

	R = {(1,1),(1,3),(3,1),(3,3),(3,4),(4,3),(4,4)}

	S = {(1,1),(1,4),(2,2),(2,3),(2,4),(3,3),(3,4),(4,4)}


















What's a Total Order?

	1. A partial order where every pair of items is related.
	2. Either (x,y) or (y,x) is in the relation for every x,y.

Is the relation a total order?

	T = {(1,1),(2,2),(3,3),(4,4),(1,2),(1,3),(2,4),(3,4),(1,4)}