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8.1 Relations and their Properties

8.1 Relations and their Properties. Definition: A binary relation R from set A to set B is Notation xRy xRy Representations: directed graph, chart. Examples. “parent”. Functions as Relations. Relations on a Set. If A is a set, then a relation on A is a relation from A to A

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8.1 Relations and their Properties

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  1. 8.1 Relations and their Properties • Definition: A binary relation R from set A to set B is • Notation • xRy • xRy • Representations: directed graph, chart

  2. Examples • “parent”

  3. Functions as Relations

  4. Relations on a Set • If A is a set, then a relation on A is a relation from A to A • How many relations are there on a set with two elements?

  5. Number of Relations on a Set • How many relations are there on a set with n elements? 14 elements?

  6. Properties of Relations on a Set A • Reflexive • Symmetric • Antisymmetric • Transitive

  7. Examples • Determine whether the relation on the set of all integers is reflexive, symmetric, antisymmetric, and/or transitive, where if and only if • a) • b)

  8. Examples • Determine whether the relation on the set of all Web pages is reflexive, symmetric, antisymmetric, and/or transitive, where if and only if • Everyone who has visited Web page has also visited Web page • There are no common links found on both Web page and Web page .

  9. Combining two Relations • Intersection • Union • Difference • Composition

  10. Examples Let and . Find a) b) c)

  11. Powers of a Relation Rn

  12. Theorem • The relation is transitive if and only if for all positive integers n.

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