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Relations

Relations. Binary Relations. a relation between elements of two sets is a subset of their Cartesian product (of ordered pairs). Note the di fference between a relation and a function: in a relation, each a ∈ A can map to multiple elements in B. Thus, relations are generalizations of functions.

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Relations

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  1. Relations

  2. Binary Relations • a relation between elements of two sets is a subset of their Cartesian product (of ordered pairs). • Note the difference between a relation and a function: in a relation, each a ∈ A can map to multiple elements in B. Thus, relations are generalizations of functions. • If an ordered pair (a, b) ∈ R then we say that a is related to b. We may also use the notation aRb.

  3. Relations

  4. Relations (Graph View)

  5. Relations (on a set)

  6. Reflexivity

  7. Symmetry I

  8. Symmetry II

  9. Symmetric Relations

  10. Transitivity

  11. Transitivity

  12. Combining Relations

  13. Composition and Powers

  14. Power Examples

  15. Equivalence Relations

  16. Equivalence Classes I

  17. Equivalence Classes II

  18. Partitions I

  19. Partitions II

  20. Matrix Interpretation

  21. Equivalence Relations (Example-I)

  22. Equivalence Relations (Example-II)

  23. Equivalence Relations (Example-III)

  24. Equivalence Relations (Example-IV)

  25. Partial Order I

  26. Partial Order II

  27. Equivalence Relations (Example-IV)

  28. Definition

  29. Comparability

  30. Total Orders

  31. Hasse Diagram

  32. Hasse Diagram Example

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