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Discrete Structures Lecture Recap: Relations and Closures

Explore matrix and digraph representations, closures, equivalence relations, and partial orders in relation theory. Understand the concepts with examples and classifications of equivalence classes and posets.

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Discrete Structures Lecture Recap: Relations and Closures

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  1. CSE 321 Discrete Structures Winter 2008 Lecture 24 Relations

  2. Announcements • Readings • Today • 8.3 Representing Relations • 8.4 Closures (Key idea – transitive closure) • 8.5 Equivalence Relations (Skim) • 8.6 Partial Orders • Next week • Graph theory

  3. Highlights from Lecture 23 • Digraph representation of relations • Matrix representation of relations

  4. Matrix multiplication Standard (, +) matrix multiplication. A is a m  n matrix, B is a n  p matrix C = A  B is a m  p matrix defined:

  5. And-OR Matrix multiplication A is a m  n boolean matrix, B is a n  p boolean matrix C = A  B is a m  p matrix defined:

  6. Matrices and Composition MS R = MR MS R = {(a, a), (a, c), (b, a), (b, b)} S = {(b, a), (b, c), (c, a), (c, c)}

  7. Closures • Reflexive Closure • Symmetric Closure

  8. Transitive Closure • R = {(1, 2), (2, 3), (3, 4)}

  9. Transitive closure

  10. Equivalence Relations Definition: A relation on a set A is called an equivalence relation if it is reflexive, symmetric, and transitive. Are these equivalence relations? • Congruence Mod m on Z+. R = {(a,b) | a  b mod m} • The ‘divides’ relation on Z+. R = {(a,b) | a|b}

  11. Equivalence classes • R = {(a,b) | a  b mod 3}, Domain: Z+

  12. Partial Orderings Definition: A relation R on a set S is called a partial ordering if it is reflexive, antisymmetric, and transitive. A set S together with a partial ordering R is called a partially ordered set, or poset. Are these posets? • (Z, ≥) • (Z+, |)

  13. Total Orderings Definition: If (S, R) is a poset and every two elements of S are comparable, S is called a totally (linearly) ordered set, and R is called a total (linear) order. Are these posets totally ordered? • (Z, ≥) • (Z+, |)

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