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Partial orderingsA relation R on a set S is called a partial ordering or partial order if it is reflexive, antisymmetric and transitive. A set S together with a partial ordering R is called a partially ordered set, or poset, and is denoted by (S, R). Example. Show that the “greater than or equal” relation ≥ is a partial ordering on the set of integers. Solution: Since Example. The divisibility relation | is a partial ordering on the set of positive integers, since it is reflexive, antisymmetric and transitive. We see that Example. Show that the inclusion relation Í is a partial ordering on the power set of a set S. Solution: Since In a poset the notation The elements a and b of a poset Example. In the poset Solution: The integers 3 and 9 are comparable, since 3 | 9. The integers 5 and 7 are incomparable, because The adjective “partial” is used to describe partial orderings since pairs of elements may be incomparable. When every two elements in the set are comparable, the relation is called a total ordering. If Example. The poset Example. The poset
Example. The set of ordered pairs of positive integers,
Date: 2015-01-02; view: 1618
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