1. Lecture 17
Relations continued
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2. Equivalence Relations
If a relation is:
Reflexive (a,a) ε R V a ε A
Symmetric a R b then b R a
Transitive a R b and b R c then a R c
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3. Equivalence Relations
Example:
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4. Equivalence Relations and
Partitions
If P is a partition of A, then P can be used
to construct an equivalence relation on
A
Theorem
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5. Equivalence Relations and
Partitions
Example
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6. Equivalence Relations and
Partitions
Proof of previous theorem
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7. Exercises
Example
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8. Manipulation of Relations
Complementary Relation
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9. Manipulation of Relations
Inverse Relation
R=(a,b)
R-1=(b,a)
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10. Exercises
Intersection, Union and Inverse
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11. Manipulation of Relations
Example
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12. Manipulation of Relations
Example
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13. Closures
Example A={a,b,c,d} and
R={(a,b),(b,c),(a,c),(c,d)}
Is this symmetric?
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14. Closures
Example
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15. Closures
Example
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16. Closures
Closures can be for any other property
such as transitive clusters, Asymmetric
antisymmetric etc
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17. Exercises
Example
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18. Exercises
Example
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