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Partial Orders http://localhost/~senning/courses/ma229/slides/partial-orders/slide01.html 1 of 1 09/11/2003 03:52 PM prev | slides | next
Partial Orders
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Partial Orders http://localhost/~senning/courses/ma229/slides/partial-orders/slide02.html 1 of 1 09/11/2003 03:52 PM
Partial Orders
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A relation R on a set S is a partial ordering or partial order if it is reflexive, antisymmetric and transitive. A set S together with a partial order R is called a partially ordered set or poset and is denoted (S,R). For example, the relation "less than or equal to" is a partial ordering
- n the integers. In this case (a,b) R if a
- b.
a
- a so R is reflexive.
a
- b implies that b
- a only when a = b so R is
antisymmetric. if a
- b and b
- c then a
- c so R is transitive.
So (Z,
- ) is a poset.
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Partial Orders http://localhost/~senning/courses/ma229/slides/partial-orders/slide03.html 1 of 1 09/11/2003 03:52 PM
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Show that the divisibility relation is a partial ordering on Z+: (a, b) R if a|b Need to show that R is reflexive, antisymmetric, transitive.
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Partial Orders http://localhost/~senning/courses/ma229/slides/partial-orders/slide04.html 1 of 1 09/11/2003 03:52 PM
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It is important to note that while this partial ordering is on Z+, there are pairs of elements a and b from Z+ such that neither (a,b) nor (b,a) are in R, the divisibility relation. 3 does not divide 5, nor does 5 divide 3. The elements a and b of a poset (S,R) are called comparable if either (a,b) R or (b,a) R. When a and b are elements of S such that neither ordered pair of them is in R then a and b are called incomparable.
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