Relations
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.
Relations
Relations (Graph View)
Relations (on a set)
Reflexivity
Symmetry I
Symmetry II
Symmetric Relations
Transitivity
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