Relation

1
Chapter 8

• Relation

2
Sec 8.1

• Relation and Their Properties

3
Definitions

• Binary Relation from A to B Let A and B be
  sets. A binary relation from A to B is a subset
  of A x B.
• Notation Let set R be a relation. If (a,b)?? R
  we write a R b.
• Relation on a set A A relation on a set A is a
  binary relation from A to A.

4
Reflexive, Symmetric, Transitive

• Reflexive A relation R on a set A is reflexive
  if (a,a) ? R whenever a ? A.
• Symmetric A relation R on a set A is symmetric
  is (b,a) ? R whenever (a,b) ? R
• Antisymmetric A relation R on a set A is
  antisymmetric is (a,b) ? R and (b,a) ? R does not
  occur except possibly when a b.
• Transitive A relation R on a set A is called
  transitive if whenever (a,b) ? R and (b,c) ? R
  then (a,c) ? R.

5
Combining Relations

• R ? S
• R ? S
• R - S
• S - R
• S ? R Let R be a relation from A to B and let S
  be a relation from B to C. The composite of R
  and S is the relation consisting of ordered pairs
  (a,c), where a ? A, c ? C, and there exists an
  element b ? B such that (a,b) ? R and (b,c) ? S.
• Rn If n 1, R1 R and if n gt 1, then Rn Rn-1
  ? R

6
Theorems

• The relation R on set A is transitive if and only
  if Rn ? R for n 1, 2, 3,

7
Homework

• Sec 8.1
• pg. 527 2, 3, 4, 5, 7, 8

8
Sec 8.3

• Representing Relations

9
Representations using Matrices

• For any Relation R let MR mij where mij 1
  if (ai,bj) ? R, and mij 0 if (ai,bj) ? R
• Reinterpreting Properties
• Reflexive mii 1 for all I
• Symmetric If mij 1 then mji 1
• Antisymmetric If mij 1 then mji 0
• Transitive MR2 ( MR2) ? MR

10
Representation using Digraphs

• A directed graph, or digraph (V,E), consists of a
  set V of vertices together with a set E of
  ordered pairs of elements of V called edges (or
  arcs). The vertex a is called the initial vertex
  of the edge (a,b), and the vertex b is called the
  terminal vertex of this edge.
• Loop an edge of the form (a,a)

11
Relation Properties

• Reflexive Ever vertex has a loop
• Symmetric Every arc in the graph has an arc
  with reversed initial and terminal endpoints
• Antisymmetric No arc in the graph has an arc
  with reversed initial and terminal endpoints
• Transitive For any two arcs in the graph there
  is a third arc which completes the cycle (A cycle
  is set of arcs that start and end at the same
  point)

12
Homework

• Sec 8.3
• pg. 542 1, 3, 4abc, 15, 23, 24, 25, 26, 27,
  31(except irreflexive)

13
Sec 8.5

• Equivalence Relations

14
Definitions

• Equivalence Relation A relation on a set A is
  called an equivalence relation if it is
  reflexive, symmetric, and transitive.
• Equivalent Two elements a and b that are
  related by an equivalence relation are called
  equivalent. The notation a?? b is used to denote
  that a and b are equivalent elements with respect
  of a particular equivalence relation.

15
Definitions

• Equivalence Classes Let R be an equivalence
  relation on a set A. The set of all elements
  that are related to an element a of A is called
  the equivalence class of a. The equivalence
  class of a with respect to R is denoted by aR
  or just a (when only one relation is under
  consideration)

16
Results

• Theorem Let R be an equivalence relation on a
  set A. The following statements are equivalent
• a ? b
• a b
• a ? b ?? ?
• In particular a ? b if and only if a b
  anda ?is not equivalent to b iff a ? b ?.

17
Results

• Theorem Let R be an equivalence relation on a
  set A. Then the equivalence classes of R form a
  partition of S.
• Conversely, given any partition of S, there is
  an equivalence relation R that has the sets Ai, i
  ? I as its equivalence classes.

18
Homework

• Sec 8.5
• pg. 562 1, 2, 3, 7, 9, 13, 21, 26, 41, 47
• End Chapter 8