Relations. Binary Relations a relation between elements of two sets is a subset of their Cartesian product (of ordered pairs). Note the di ﬀ erence between.

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Presentation transcript:

Relations

Binary Relations a relation between elements of two sets is a subset of their Cartesian product (of ordered pairs). Note the di ﬀ erence 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.

Relations

Relations (Graph View)

Relations (on a set)

Reflexivity

Symmetry I

Symmetry II

Symmetric Relations

Transitivity

Combining Relations

Composition and Powers

Power Examples

Equivalence Relations

Equivalence Classes I

Equivalence Classes II

Partitions I

Partitions II

Matrix Interpretation

Equivalence Relations (Example-I)

Equivalence Relations (Example-II)

Equivalence Relations (Example-III)

Equivalence Relations (Example-IV)

Partial Order I

Partial Order II

Equivalence Relations (Example-IV)

Definition

Comparability

Total Orders

Hasse Diagram

Hasse Diagram Example