Relations

1
Relations
2
Binary Relations

• a relation between elements of two sets is a
  subset of their Cartesian product (of ordered
  pairs).
• Note the di?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.

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