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8.1 Relations and their Properties

This comprehensive guide explores binary relations defined from set A to set B, including key notations and various representations such as directed graphs and charts. We analyze common relations like "parent" and discuss properties such as reflexivity, symmetry, antisymmetry, and transitivity, providing concrete examples for clarity. Further, we investigate the number of possible relations on sets of different sizes and demonstrate how to combine relations through intersection, union, difference, and composition. Practical applications of these concepts provide deeper insights into relational structures.

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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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