Equivalence Relations

1
Lecture 2

• Equivalence Relations
• Reading Epp Chp 10.3

2
Overview Equivalence Relations

• Revision
• Definition of an Equivalence Relation
• Examples (and non-examples)
• Visualization Tool
• From Equivalence Relations to Equivalence
  Classes to Partitions
• From Partitions to Equivalence Relations
• Another Example

3
1. Revision
Concrete World
Relation R from A to B
Abstract World
Q What happens if A B?
4
1. Revision
Concrete World
Relation R on A
Everyone is related to himself
If x is related to y and y is related to z, then
x is related to z.
If x is related to y, then y is related to x
If x is related to y and y is related to x, then
x y.
5
1. Revision

• Given a relation R on a set A,
• R is reflexive iff
• "xÎA, x R x
• R is symmetric iff
• "x,yÎA, x R y y R x
• R is anti-symmetric iff
• "x,yÎA, x R y Ù y R x xy
• R is transitive iff
• "x,yÎA, x R y Ù y R z x R z

6
Overview Equivalence Relations

• Revision
• Definition of an Equivalence Relation
• Examples (and non-examples)
• Visualization Tool
• From Equivalence Relations to Equivalence
  Classes to Partitions
• From Partitions to Equivalence Relations
• Another Example

7
2. Definition

• Given a relation R on a set A,
• R is an equivalence relation iff
• R is reflexive, symmetric and transitive.
  (Todays Lecture)
• R is a partial order iff
• R is reflexive, anti-symmetric and transitive.
• (Next Lectures)

8
2. Definition

• Given a relation R on a set A,
• R is an equivalence relation iff
• R is reflexive, symmetric and transitive.

• Q How do I check whether a relation is an
  equivalence relation?
• A Just check whether it is reflexive, symmetric
  and transitive. (Always go back to the
  definition.)
• Q How do I check whether a relation is
  reflexive, symmetric and transitive?
• A Again, go back to the definitions of
  reflexive, symmetric and transitive. (Previous
  Lecture)

9
3 Examples (EqRel in life)

• 3.1 Let S be the set of all second year students.
  Define a relation C on S such that
• x C y iff x and y take at least 1 course in
  common

Q1 Is C reflexive? ("xÎS, x C x) ??? Yes. Q2
Is C symmetric? ("x,yÎS, x C y y C x) ???
Yes. Q3 Is C transitive? ("x,yÎS, x C y Ù y
C z x C z) ??? NO!!! Therefore C is NOT an
equivalence relation.
10
3 Examples (EqRel in life)

• 3.2 Let S be the set of all second year students.
  Define a relation N on S such that
• x N y iff x and y take NO courses in common

Q1 Is N reflexive? ("xÎS, x N x) ???
NO!!!. Q2 Is N symmetric? ("x,yÎS, x N y
y N x) ??? Yes. Q3 Is N transitive?
("x,yÎS, x N y Ù y N z x N z) ???
NO!!! Therefore N is NOT an equivalence relation.
11
3 Examples (EqRel in life)

• 3.3 Let S be the set of all people this room.
  Define a relation T on S such that
• x T y iff x is of equal or taller height than y

Q1 Is T reflexive? ("xÎS, x T x) ??? Yes. Q2
Is T symmetric? ("x,yÎS, x T y y T x) ???
NO!!! Q3 Is T transitive? ("x,yÎS, x T y Ù y
T z x T z) ??? Yes. Therefore T is NOT an
equivalence relation.
12
3 Examples (EqRel in life)

• 3.4 Let S be the set of all people in this room.
  Define a relation M on S such that
• x M y iff x is born in the same month as y

Q1 Is M reflexive? ("xÎS, x M x) ??? Yes. Q2
Is M symmetric? ("x,yÎS, x M y y M x) ???
Yes. Q3 Is M transitive? ("x,yÎS, x M y Ù y
M z x M z) ??? Yes. Therefore M is an
equivalence relation.
13
3 Examples (Finite Eq Rels)

• 3.5 Let A 0,1,2,3,4
• Let R (0,0), (0,4), (1,1), (1,3), (2,2),
  (4,0), (3,3), (3,1), (4,4)
• Is R an equivalence relation?

• Q1 Is R reflexive?
• Reflexive "xÎA, x R x (Always go back to
  the definition)
• Yes!

14
3 Examples (Finite Eq Rels)

• 3.5 Let A 0,1,2,3,4
• Let R (0,0), (0,4), (1,1), (1,3), (2,2),
  (4,0), (3,3), (3,1), (4,4)
• Is R an equivalence relation?

• Q2 Is R symmetric?
• Symmetric "x,yÎA, x R y y R x (Always go
  back to the definition)

15
3 Examples (Finite Eq Rels)

• 3.5 Let A 0,1,2,3,4
• Let R (0,0), (0,4), (1,1), (1,3), (2,2),
  (4,0), (3,3), (3,1), (4,4)
• Is R an equivalence relation?

• Q2 Is R symmetric?
• Symmetric "x,yÎA, x R y y R x (Always go
  back to the definition)

16
3 Examples (Finite Eq Rels)

• 3.5 Let A 0,1,2,3,4
• Let R (0,0), (0,4), (1,1), (1,3), (2,2),
  (4,0), (3,3), (3,1), (4,4)
• Is R an equivalence relation?

• Q2 Is R symmetric?
• Symmetric "x,yÎA, x R y y R x (Always go
  back to the definition)

17
3 Examples (Finite Eq Rels)

• 3.5 Let A 0,1,2,3,4
• Let R (0,0), (0,4), (1,1), (1,3), (2,2),
  (4,0), (3,3), (3,1), (4,4)
• Is R an equivalence relation?

• Q2 Is R symmetric?
• Symmetric "x,yÎA, x R y y R x (Always go
  back to the definition)

18
3 Examples (Finite Eq Rels)

• 3.5 Let A 0,1,2,3,4
• Let R (0,0), (0,4), (1,1), (1,3), (2,2),
  (4,0), (3,3), (3,1), (4,4)
• Is R an equivalence relation?

• Q2 Is R symmetric?
• Symmetric "x,yÎA, x R y y R x (Always go
  back to the definition)

19
3 Examples (Finite Eq Rels)

• 3.5 Let A 0,1,2,3,4
• Let R (0,0), (0,4), (1,1), (1,3), (2,2),
  (4,0), (3,3), (3,1), (4,4)
• Is R an equivalence relation?

• Q2 Is R symmetric?
• Symmetric "x,yÎA, x R y y R x (Always go
  back to the definition)
• Yes, R is symmetric.

20
3 Examples (Finite Eq Rels)

• 3.5 Let A 0,1,2,3,4
• Let R (0,0), (0,4), (1,1), (1,3), (2,2),
  (4,0), (3,3), (3,1), (4,4)
• Is R an equivalence relation?

• Q3 Is R transitive?
• Transitive "x,yÎA, x R y Ù y R z x R z
  (Always go back to the definition)

21
3 Examples (Finite Eq Rels)

• 3.5 Let A 0,1,2,3,4
• Let R (0,0), (0,4), (1,1), (1,3), (2,2),
  (4,0), (3,3), (3,1), (4,4)
• Is R an equivalence relation?

• Q3 Is R transitive?
• Transitive "x,yÎA, x R y Ù y R z x R z
  (Always go back to the definition)

22
3 Examples (Finite Eq Rels)

• 3.5 Let A 0,1,2,3,4
• Let R (0,0), (0,4), (1,1), (1,3), (2,2),
  (4,0), (3,3), (3,1), (4,4)
• Is R an equivalence relation?

• Q3 Is R transitive?
• Transitive "x,yÎA, x R y Ù y R z x R z
  (Always go back to the definition)

23
3 Examples (Finite Eq Rels)

• 3.5 Let A 0,1,2,3,4
• Let R (0,0), (0,4), (1,1), (1,3), (2,2),
  (4,0), (3,3), (3,1), (4,4)
• Is R an equivalence relation?

• Q3 Is R transitive?
• Transitive "x,yÎA, x R y Ù y R z x R z
  (Always go back to the definition)

24
3 Examples (Finite Eq Rels)

• 3.5 Let A 0,1,2,3,4
• Let R (0,0), (0,4), (1,1), (1,3), (2,2),
  (4,0), (3,3), (3,1), (4,4)
• Is R an equivalence relation?

• Q3 Is R transitive?
• Transitive "x,yÎA, x R y Ù y R z x R z
  (Always go back to the definition)
• Carry on with checking
• Yes, R is transitive.

25
3 Examples (Infinite Eq Rels)

• 3.6 Let R be a relation on Z, such that
• x R y iff x º y (mod 3)
• Prove that R an equivalence relation.

• NOTE Do not be confuse with the º notation
• GO BACK TO THE DEFINITION
• x º y (mod k) iff k (x-y)
• Remember? x is congruent to y modulo k?
• (Meaning that x and y will give the same
  remainder when divided by k)

• Therefore, by definition
• x º y (mod 3) iff 3 (x-y)

26
3 Examples (Infinite Eq Rels)

• 3.6 Let R be a relation on Z, such that
• x R y iff x º y (mod 3)
• Prove that R an equivalence relation.

• To show that R is Reflexive
• We must show x º x (mod 3)
• Hence, we must show 3 (x - x)
• Which is true, since 3 0.
• Hence R is Reflexive.

27
3 Examples (Infinite Eq Rels)

• 3.6 Let R be a relation on Z, such that
• x R y iff x º y (mod 3)
• Prove that R an equivalence relation.

• To show that R is Symmetric
• Assume x R y
• x º y (mod 3) (Defn of R)
• 3 (x - y) (Defn of congruence modulo)
• 3 (y - x) (DM1 Thm3.2.4 mn m-n)
• y º x (mod 3) (Defn of congruence modulo)
• y R x (Defn of R)

28
3 Examples (Infinite Eq Rels)

• 3.6 Let R be a relation on Z, such that
• x R y iff x º y (mod 3)
• Prove that R an equivalence relation.

• To show that R is Transitive
• Assume x R y and y R z.
• x º y (mod 3) and y º z (mod 3) (Defn of R)
• 3 (x - y) and 3 (y - z) (Defn of º mod)
• (x - y) 3j and (y - z) 3k (Defn of )
• (x - z) 3(jk) (Add both eqns)
• 3 (x - z) (Defn of )
• x º z (mod 3) (Defn of º mod)
• x R z (Defn of R)

29
3 Examples (Infinite Eq Rels)

• 3.6 Let R be a relation on Z, such that
• x R y iff x º y (mod 3)

Q How does an equivalence relation look
like? Well, lets draw the above equivalence
relation and see But first, before we draw, we
must find out some of the numbers which are
related.
30
3 Examples (Infinite Eq Rels)

• 3.6 Let R be a relation on Z, such that
• x R y iff x º y (mod 3)

• DEFINITION x º y (mod 3) iff 3 (x-y)

0 R 0 since 0 º 0 (mod 3) 0 R 3 since 0 º 3
(mod 3) 3 R 0 since 3 º 0 (mod 3) 3 R 3 since
3 º 3 (mod 3) 0 R 6 since 0 º 6 (mod 3) 6 R 0
since 6 º 0 (mod 3) 3 R 6 since 3 º 6 (mod 3)
6 R 3 since 6 º 3 (mod 3) 9 R 9, 0 R 9, 9 R
0, 3 R 9, 9 R 3 6 R 9, 9 R 6,
0 R -3 since 0 º -3 (mod 3) -3 R 0 since -3 º 0
(mod 3) -3 R -3 since -3 º -3 (mod 3) 0 R
-6 since 0 º -6 (mod 3) -6 R 0 since -6 º 0
(mod 3) -6 R -6 since -6 º -6 (mod 3) -3 R
-6 since -3 º -6 (mod 3) -6 R -3 since -6 º -3
(mod 3) -9 R -9, 0 R -9, -9 R 0, -3 R -9, -9 R
-3 -6 R -9, -9 R -6,
31
3 Examples (Infinite Eq Rels)

• 3.6 Let R be a relation on Z, such that
• x R y iff x º y (mod 3)

• DEFINITION x º y (mod 3) iff 3 (x-y)

1 R 1 since 1 º 1 (mod 3) 1 R 4 since 1 º 4
(mod 3) 4 R 1 since 4 º 1 (mod 3) 4 R 4 since
4 º 4 (mod 3) 1 R 7 since 1 º 7 (mod 3) 7 R 1
since 7 º 1 (mod 3) 4 R 7 since 4 º 7 (mod 3)
7 R 4 since 7 º 4 (mod 3)
1 R -2 since 1 º -2 (mod 3) -2 R 1 since -2 º 1
(mod 3) -2 R -2 since -2 º -2 (mod 3) 1 R
-5 since 1 º -5 (mod 3) -5 R 1 since -5 º 1
(mod 3) -5 R -5 since -5 º -5 (mod 3) -2 R
-5 since -2 º -5 (mod 3) -5 R -2 since -5 º -2
(mod 3)
32
3 Examples (Infinite Eq Rels)

• 3.6 Let R be a relation on Z, such that
• x R y iff x º y (mod 3)

• DEFINITION x º y (mod 3) iff 3 (x-y)

2 R 2 since 2 º 2 (mod 3) 2 R 5 since 2 º 5
(mod 3) 5 R 2 since 5 º 2 (mod 3) 5 R 5 since
5 º 5 (mod 3) 2 R 8 since 2 º 8 (mod 3) 8 R 2
since 8 º 2 (mod 3) 5 R 8 since 5 º 8 (mod 3)
8 R 5 since 8 º 5 (mod 3)
2 R -1 since 2 º -1 (mod 3) -1 R 2 since -1 º 2
(mod 3) -1 R -1 since -1 º -1 (mod 3) 2 R
-4 since 2 º -4 (mod 3) -4 R 2 since -4 º 2
(mod 3) -4 R -4 since -4 º -4 (mod 3) -1 R
-4 since -1 º -4 (mod 3) -4 R -1 since -4 º -1
(mod 3)
33
3 Examples (Infinite Eq Rels)

• 3.6 Let R be a relation on Z, such that
• x R y iff x º y (mod 3)

• DEFINITION x º y (mod 3) iff 3 (x-y)

2 R 2 2 R 5 5 R 2 5 R 5 2 R 8 8 R 2 5 R 8 8 R 5
2 R -1 -1 R 2 -1 R -1 2 R -4 -4 R 2 -4 R -4 -1 R
-4 -4 R -1
1 R 1 1 R 4 4 R 1 4 R 4 1 R 7 7 R 1 4 R 7 7 R 4
1 R -2 -2 R 1 -2 R -2 1 R -5 -5 R 1 -5 R -5 -2 R
-5 -5 R -2
0 R 0 0 R 3 3 R 0 3 R 3 0 R 6 6 R 0 3 R 6 6 R 3
0 R -3 -3 R 0 -3 R -3 0 R -6 -6 R 0 -6 R -6 -3 R
-6 -6 R -3
34
4. Visual Tool RÍZ2, x R y iff xºy (mod 3)
0
3
-3
6
9
R (0,0),
(3,3)
,(0,3)
,(3,0)
,(6,6)
,(0,6)
,(6,0)
,(3,6)
,(6,3)
,(9,9)
,(0,9)
,(9,0)
,(3,9)
,(9,3)
,(6,9)
,(9,6)
,(-3,-3)
,(0,-3)
,(-3,0)
,(3,-3)
,(3-,3)
,(6,-3)
,(-3,6)
,(9,-3)
,(-3,9)
..
35
4. Visual Tool RÍZ2, x R y iff xºy (mod 3)
0
1
3
4
-2
-3
6
7
9
10
36
4. Visual Tool RÍZ2, x R y iff xºy (mod 3)
0
1
3
4
-2
-3
6
7
9
10

• Notice that we will have a group of elements,
  everyone related to everyone else in that group -
  Very troublesome to draw lines everywhere since
  we know everyone is related to everyone else in
  that group.

37
4. Visual Tool RÍZ2, x R y iff xºy (mod 3)
0
1
3
4
-2
-3
6
7
9
10

• Notice also that we have a few distinct groups.
  No one from one group is related to another
  element from another group.
• Lets simplify the drawing

38
4. Visual Tool RÍZ2, x R y iff xºy (mod 3)
0
1
2
3
4
5
-2
-3
-1
6
7
8
9
10
11

• We can partition Z into a few sets based on the
  rule that the elements within a partition are
  related to each other, and the elements between
  partitions are not related to each other.

39
4. Visual Tool RÍZ2, x R y iff xºy (mod 3)
0
1
2
3
4
5
-2
-3
-1
6
7
8
9
10
11

• After partitioning, we can drop the arrows
  depicting the relationships, because we know that
  everyone within the same partition is relation to
  each other.

40
4. Visual Tool RÍZ2, x R y iff xºy (mod 3)
0
1
2
3
4
5
-2
-3
-1
6
7
8
9
10
11

• After partitioning, we can drop the arrows
  depicting the relationships, because we know that
  everyone within the same partition is relation to
  each other.

41
4. Visual Tool RÍZ2, x R y iff xºy (mod 3)
0
1
2
3
4
5
-2
-3
-1
6
7
8
9
10
11
The partitioning of Z, shown here in a diagram
can be written down as ,-6,-3,0,3,6,,,-5
,-2,1,4,7,,,-4,-1,2,5,8,
42
4. Visual Tool RÍZ2, x R y iff xºy (mod 3)
0
1
2
3
4
5
-2
-3
-1
6
7
8
9
10
11
Q Does this partitioning work for every
equivalence relation? Can every equivalence
relation R on a set A, be partitioned in this
manner? A Yes. We will now prove it in general.
43
4. Visual Tool RÍZ2, x R y iff xºy (mod 3)
Instead of writing ,-6,-3,0,3,6,, we define
the concept of an equivalence class. The idea is
that we take one element to represent all the
elements of that partition.
Definition Let R be an equivalence relation on
a set A, and let a Î A. The
equivalence class of a is defined as follows a
x Î A x R a (i.e. x Î a iff x R
a) Note (1) a is known as a representative.
(2) a is a SET
44
4. Visual Tool RÍZ2, x R y iff xºy (mod 3)
By analogy, consider the earlier example 3.4 Let
A be all the people in this room. We define a
relation R on S such that x R y iff x and
y are born in the same month (We have seen
earlier the R is an equivalence relation) A can
be divided in the twelve partitions. Previous
slide defn a x Î A x R a (i.e. x Î
a iff x R a) So what is Riza? Riza x
Î A x R Riza x Î A x and Riza are
born in the same month And what is
Mika? Mika x Î A x R Mika x Î
A x and Mika are born in the same month is
Riza Mika?
45
4. Visual Tool RÍZ2, x R y iff xºy (mod 3)
So similarly, in this congruence modulo relation
a x Î A x R a (i.e. x Î a iff x R a)
Q What is 0? A By Definition 0 x Î Z
x R 0 x Î Z x º 0 (mod 3) 0
,-3,0,3,6,9,
Q What is 3? A By Definition 3 x Î Z
x R 3 x Î Z x º 0 (mod 3) 3
,-3,0,3,6,9,
46
4. Visual Tool RÍZ2, x R y iff xºy (mod 3)
So similarly, in this congruence modulo relation
a x Î A x R a (i.e. x Î a iff x R a)
0 3 ,-3,0,3,6,9, In fact, 0 3
-3 6 -6 ,-3,0,3,6,9, We just need
one representative of that class.
47
4. Visual Tool RÍZ2, x R y iff xºy (mod 3)
So similarly, in this congruence modulo relation
a x Î A x R a (i.e. x Î a iff x R a)
Q What is 1? A By Definition 1 x Î Z
x R 1 x Î Z x º 1 (mod 3) 1
,-5,-2,1,4,7,10,13,16
1 -2 4 7 ,-5,-2,1,4,7,10,13,16
All refer to the same partition.
48
4. Visual Tool RÍZ2, x R y iff xºy (mod 3)
So similarly, in this congruence modulo relation
a x Î A x R a (i.e. x Î a iff x R a)
Q What is 8? A By Definition 8 x Î Z
x R 8 x Î Z x º 8 (mod 3) 8
,-4,-1,2,5,8,11,14,17
-1 2 5 8 ,-4,-1,2,5,8,11,14,17
All refer to the same partition.
49
4. Visual Tool RÍZ2, x R y iff xºy (mod 3)
We make a guess on the following generalisations
(1) Given an equivalence relation R on A, and a,b
Î A If a R b then a b
(2) Given an equivalence relation R on A, and a,b
Î A Either a Ç b f or a b
(3) Given an equivalence relation R on A, the
distinct equivalence classes of R form a
partition of A.
50
5. From Eq Rel to Eq Class to Partitions
Equivalence Relation

• Prove Given an equivalence relation R on A, and
  a,b Î A
• If a R b then a b
• Either a Ç b f or a b
• The distinct equivalence classes of R form a
  partition of A.

Partition
51
5.1 Proof

• (1) Given an equivalence relation R on A, and a,b
  Î A
• If a R b then a b

Proof a Í b Assume e Î a e R a (by
definition x Î a iff x R a ) e R a Ù a R b
(since we assumed that a R b) e R b (Since
R, being an Eq Rel, is transitive) e Î b
(by definition x Î b iff x R b ) Therefore
a Í b (Note We cant flip the arrow and go
backwards)
52
5.1 Proof

• (1) Given an equivalence relation R on A, and a,b
  Î A
• If a R b then a b

Proof b Í a Assume e Î b e R b (by
definition x Î b iff x R b ) e R b Ù a R b
(since we assumed that a R b) e R b Ù b R a
(Since R, being an Eq Rel, is symmetric) e R
a (Since R, being an Eq Rel, is transitive) e
Î a (by definition x Î b iff x R b
) Therefore b Í a
53
5.2 Proof

• (2) Given an equivalence relation R on A, and a,b
  Î A
• Either a Ç b f or a b

Proof Either a Ç b f or a Ç b ¹
f Case 1 Assuming that a Ç b f, then we
are done! Since its true that a Ç b f
or a b Case 2 Assuming that a Ç b
¹ f next slide a Ç b f or a
b
54
5.2 Proof
e Î a Ç b (Defn of f ("x, x Ï S)
Sf) e Î a Ù e Î b (By defn of Ç) e R
a Ù e R b (By defn of Eq Class) a R e Ù e
R b (Since R, an Eq Rel is symmetric) a R
b (Since R, an Eq Rel is transitive) a
b (Proven IF a R b THEN a b) Proven.
55
5.2 Proof
Lets sidetrack a bit and take a closer look at
the proof
Assuming that a Ç b ¹ f then e Î a Ç
b (Defn of f ("x, x Ï S) Sf) e Î a
Ù e Î b (By defn of Ç) e R a Ù e R
b (By defn of Eq Class) a R e Ù e R
b (Since R, an Eq Rel is symmetric) a R
b (Since R, an Eq Rel is transitive) a
b (Proven IF a R b THEN a b) a Ç
b f or a b
Look here Do you know that a lot of things are
happening here?
56
5.2 Proof
Lets sidetrack a bit and take a closer look at
the proof
Assuming that a Ç b ¹ f then e Î a Ç
b (Defn of f ("x, x Ï S) Sf) e Î a
Ù e Î b (By defn of Ç) e R a Ù e R
b (By defn of Eq Class) a R e Ù e R
b (Since R, an Eq Rel is symmetric) a R
b (Since R, an Eq Rel is transitive) a
b (Proven IF a R b THEN a b) a Ç
b f or a b
57
5.2 Proof
Lets sidetrack a bit and take a closer look at
the proof
Assuming that a Ç b ¹ f then (Defn of f
("x, x Ï S) Sf) e Î a Ç b e Î a
Ù e Î b (By defn of Ç) e R a Ù e R
b (By defn of Eq Class) a R e Ù e R
b (Since R, an Eq Rel is symmetric) a R
b (Since R, an Eq Rel is transitive) a
b (Proven IF a R b THEN a b) a Ç
b f or a b
58
5.2 Proof
Lets sidetrack a bit and take a closer look at
the proof
Assuming that a Ç b ¹ f then (Defn of f
("x, x Ï S) Sf) e Î a Ç b e Î a
Ù e Î b (By defn of Ç) e R a Ù e R
b (By defn of Eq Class) a R e Ù e R
b (Since R, an Eq Rel is symmetric) a R
b (Since R, an Eq Rel is transitive) a
b (Proven IF a R b THEN a b) a Ç
b f or a b a Ç b f or a b
59
5.2 Proof
Lets sidetrack a bit and take a closer look at
the proof
Assuming that a Ç b ¹ f then (Defn of f
("x, x Ï S) Sf) e Î a Ç b e Î
a Ù e Î b (By defn of Ç) e R a Ù e R
b (By defn of Eq Class) a R e Ù e R
b (Since R, an Eq Rel is symmetric) a R
b (Since R, an Eq Rel is transitive) a
b (Proven IF a R b THEN a b) a Ç
b f or a b a Ç b f or a b
60
5.2 Proof
Lets sidetrack a bit and take a closer look at
the proof
Assuming that a Ç b ¹ f then x, (x Î a Ç
b) (Defn of f ("x, x Ï S) Sf) e Î a
Ç b e Î a Ù e Î b (By defn of
Ç) e R a Ù e R b (By defn of Eq Class) a
R e Ù e R b (Since R, an Eq Rel is
symmetric) a R b (Since R, an Eq Rel is
transitive) a b (Proven IF a R b THEN
a b) a Ç b f or a b a Ç
b f or a b
61
5.2 Proof
Lets sidetrack a bit and take a closer look at
the proof
Assuming that a Ç b ¹ f then x, (x Î a Ç
b) (Defn of f ("x, x Ï S) Sf) e Î a
Ç b e Î a Ù e Î b (By defn of
Ç) e R a Ù e R b (By defn of Eq Class) a R
e Ù e R b (Since R, an Eq Rel is
symmetric) a R b (Since R, an Eq Rel is
transitive) a b (Proven IF a R b THEN
a b) a Ç b f or a b a Ç
b f or a b
Looks familiar?
62
5.2 Proof
Lets sidetrack a bit and take a closer look at
the proof
Assuming that a Ç b ¹ f then x, (x Î a Ç
b) (Defn of f ("x, x Ï S) Sf) e Î a
Ç b e Î a Ù e Î b (By defn of
Ç) e R a Ù e R b (By defn of Eq Class) a
R e Ù e R b (Since R, an Eq Rel is
symmetric) a R b (Since R, an Eq Rel is
transitive) a b (Proven IF a R b THEN
a b) a Ç b f or a b a Ç
b f or a b
63
5.2 Proof
Lets sidetrack a bit and take a closer look at
the proof
Assuming that a Ç b ¹ f then (Defn of f
("x, x Ï S) Sf) e Î a Ç b e Î
a Ù e Î b (By defn of Ç) e R a Ù e R
b (By defn of Eq Class) a R e Ù e R
b (Since R, an Eq Rel is symmetric) a R
b (Since R, an Eq Rel is transitive) a
b (Proven IF a R b THEN a b) a Ç
b f or a b
64
5.2 Proof
Lets sidetrack a bit and take a closer look at
the proof
Assuming that a Ç b ¹ f then (Defn of f
("x, x Ï S) Sf) e Î a Ç b e Î a
Ù e Î b (By defn of Ç) e R a Ù e R
b (By defn of Eq Class) a R e Ù e R
b (Since R, an Eq Rel is symmetric) a R
b (Since R, an Eq Rel is transitive) a
b (Proven IF a R b THEN a b) a Ç
b f or a b
65
5.2 Proof
Lets sidetrack a bit and take a closer look at
the proof
Assuming that a Ç b ¹ f then e Î a Ç
b (Defn of f ("x, x Ï S) Sf) e Î a
Ù e Î b (By defn of Ç) e R a Ù e R
b (By defn of Eq Class) a R e Ù e R
b (Since R, an Eq Rel is symmetric) a R
b (Since R, an Eq Rel is transitive) a
b (Proven IF a R b THEN a b) a Ç
b f or a b
Although you dropped certain symbols, be aware
that they are still very much there.
66
5. From Eq Rel to Eq Class to Partitions
Equivalence Relation

• Prove Given an equivalence relation R on A, and
  a,b Î A
• If a R b then a b
• Either a Ç b f or a b
• The distinct equivalence classes of R form a
  partition of A.

Partition
67
5.3 Proof

• (3) Given an equivalence relation R on A, the
  distinct equivalence classes of R form a
  partition of A.

Proof Let A1, A2, ,An be distinct equivalence
classes of R. To do Show that A1, A2, ,An form
a partition of A. Q What is the definition of a
partition? A A1, A2, ,An is a partition of A
iff (1) A A1 È A2 È È An (2) "i,j, i ¹ j
Ai Ç Aj f (Mutually disjoint)
68
5.3 Proof

• (3) Given an equivalence relation R on A, the
  distinct equivalence classes of R form a
  partition of A.

Proof Let A1, A2, ,An be distinct equivalence
classes of R. To do Show that (1) A A1 È A2
È È An (2) "i,j, i ¹ j Ai Ç Aj f (Mutually
disjoint)
69
5.3 Proof

• (3) Given an equivalence relation R on A, the
  distinct equivalence classes of R form a
  partition of A.

Proof (1) Show that A A1 È A2 È È An
(a) Show A Í A1 È A2 È È An (b) Show A1
È A2 È È An Í A
70
5.3 Proof

• (3) Given an equivalence relation R on A, the
  distinct equivalence classes of R form a
  partition of A.

Proof (1) Show that A A1 È A2 È È An
(a) Show A Í A1 È A2 È È An
Let x Î A x R x (Since R, an Eq Rel, is
reflexive) x Î x (By defn of Eq Class) x
must be one of the Ai (i from 1 to n) x Î Ai
(for some i from 1 to n) x Î A1 È A2 È È An
71
5.3 Proof

• (3) Given an equivalence relation R on A, the
  distinct equivalence classes of R form a
  partition of A.

Proof (1) Show that A A1 È A2 È È An
(a) Show A Í A1 È A2 È È An (b) Show A1
È A2 È È An Í A
Let x Î A1 È A2 È È An x Î Ai (for some i
from 1 to n, by defn of Union) But each Ai is an
Eq Class of R (we assumed this) and any Eq Class
is a subset of A (Remember the Defn a x Î
A x R a ) Therefore Ai Í A. Hence x Î A
72
5.3 Proof

• (3) Given an equivalence relation R on A, the
  distinct equivalence classes of R form a
  partition of A.

Proof (2) Show that "i,j, i ¹ j Ai Ç Aj f
(Mutually disjoint) Assume i ¹ j, and so
the 2 sets Ai ¹ Aj Then there exists
elements a and b in A such that Ai a and Aj
b Now, either a Ç b f or a
b (Lemma proven) But we know a ¹ b.
Therefore a Ç b f. Hence Ai Ç Aj f
73
5.4 Where are we?
Equivalence Relation
Partition
74
6. From Partitions to Eq Rel.

• Defn Given a set A, and partition of A (A A1
  È A2 È È An), the (binary) relation R, induced
  by the partition, is defined as
• x R y iff (i, x Î Ai and y Î Ai )
• (meaning to say x R y iff x and y belong to the
  same partition)
• For example, let A 0,1,2,3,4 and let 0,3,4,
  1, 2 be a partition of A. Let R be the
  relation induced by this partition.

• Question

• Is 0 R 0?

• Is 1 R 2?

Yes.
No.

• Is 0 R 3?

• Is 3 R 4?

Yes.
Yes.

• Is 0 R 1?

• Is 1 R 1?

No.
Yes.

• List the elements in R

R (0,0),(3,3),(4,4),(0,3),(3,0),(0,4),(4,0),(3,
4),(4,3),(1,1),(2,2)
75
6. From Partitions to Eq Rel.

• A

The (binary) relation induced by the partition, R
is defined as "x,yÎA, x R y iff x and y belong
to the same partition of A
76
6. From Partitions to Eq Rel.

• (Thm) Let A be a set with a partition and let R
  be the relation induced by the partition. Then R
  is an equivalence relation. x R y iff (i, x
  Î Ai and y Î Ai)

• Proof (Reflexive)
• Pick any x in A.
• Then x Î Ai for some i. (meaning that x must be
  in some partition)
• So i, x Î Ai and x Î Ai
• So x R x.
• So R is reflexive.

77
6. From Partitions to Eq Rel.

• (Thm) Let A be a set with a partition and let R
  be the relation induced by the partition. Then R
  is an equivalence relation. x R y iff (i, x
  Î Ai and y Î Ai)

• Proof (symmetric)
• Assume x R y
• Then (i, x Î Ai and y Î Ai)
• Then (i, y Î Ai and x Î Ai)
• So y R x.

78
6. From Partitions to Eq Rel.

• (Thm) Let A be a set with a partition and let R
  be the relation induced by the partition. Then R
  is an equivalence relation. x R y iff (i, x
  Î Ai and y Î Ai)

• Proof (transitive)
• Assume x R y and y R z
• Then x Î Ai and y Î Ai (for some i) and y Î
  Ak and z Î Ak (for some k)
• Claim ik.
• If i ¹ k, then Ai Ç Aj f. (defn partitions are
  mutually disjoint). But we know that y Î Ai, and
  also that y Î Ak. Meaning that y Î Ai Ç Aj
  meaning that Ai Ç Aj ¹ f ( !)
• Therefore (i, x Î Ai and z Î Ai)
• Therefore x R z.

79
6. From Partitions to Eq Rel.
Equivalence Relation
Partition
80
7. Example

• Let R be a relation on (Z)2, such that
• (a,b) R (c,d) iff ad bc
• Is R an equivalence relation?

• R is a relation on (Z)2!!! Wow! Confusing.
• Just go back to the definition
• R is a relation on A means
• R Í (A x A) or R Í A2
• R is a relation on (Z)2 means
• R Í ((Z)2 x (Z)2) or R Í ((Z)2)2

81
7. Example

• Let R be a relation on (Z)2, such that
• (a,b) R (c,d) iff ad bc
• Is R an equivalence relation?

• Is R reflexive? Go back to definition
• If R is on A, reflexive means "xÎA, x R x
• Now, R is on (Z)2, so reflexive means
• "(m,n)Î(Z)2 , (m,n) R (m,n)
• It is NOT
• "(m,m)Î(Z)2 , (m,m) R (m,m)
• Be very careful!!!

82
7. Example

• Let R be a relation on (Z)2, such that
• (a,b) R (c,d) iff ad bc
• Is R an equivalence relation?

• Is R reflexive? Go back to definition
• "(m,n)Î(Z)2 , (m,n) R (m,n)
• The above is true in general since given any
  (m,n)
• (m,n) R (m,n) due to the fact that mn nm
• Therefore R is reflexive.

83
7. Example

• Let R be a relation on (Z)2, such that
• (a,b) R (c,d) iff ad bc
• Is R an equivalence relation?

• Is R symmetric? Go back to definition
• If R is on A, symmetric means
• "x,yÎA, x R y y R x
• Now, R is on (Z)2, so symmetric means
• "(a,b), (c,d)Î(Z)2 , (a,b) R (c,d) (c,d) R
  (a,b)

84
7. Example

• Let R be a relation on (Z)2, such that
• (a,b) R (c,d) iff ad bc
• Is R an equivalence relation?

• Is R symmetric? Go back to definition
• "(a,b), (c,d)Î(Z)2 , (a,b) R (c,d) (c,d) R
  (a,b)
• Assume (a,b) R (c,d)
• ad bc
• bc ad
• cb da
• (c,d) R (a,b)
• Yes, R is symmetric

85
7. Example

• Let R be a relation on (Z)2, such that
• (a,b) R (c,d) iff ad bc
• Is R an equivalence relation?

• Is R transitive? Go back to definition
• If R is on A, transitive means
• "x,y,zÎA, x R y Ù y R z x R z
• Now, R is on (Z)2, so transitive means
• "(a,b),(c,d),(e,f)Î(Z)2 ,
• (a,b) R (c,d) Ù (c,d) R (e,f) (a,b) R (e,f)

86
7. Example

• Let R be a relation on (Z)2, such that
• (a,b) R (c,d) iff ad bc
• Is R an equivalence relation?

• Is R transitive? Go back to definition
• "(a,b),(c,d),(e,f)Î(Z)2 ,
• (a,b) R (c,d) Ù (c,d) R (e,f) (a,b) R (e,f)
• Assume (a,b) R (c,d) and (c,d) R (e,f)
• ad bc and cf de (by definition)
• a/b c/d and c/de/f
• a/b e/f. Therefore af eb. And so af be
• Hence (a,b) R (e,f)

87
7. Example

• Let R be a relation on (Z)2, such that
• (a,b) R (c,d) iff ad bc
• What are the distinct Eq Classes of R?

• (1,1) (1,1), (2,2), (3,3), (4,4),
• (1,2) (1,2), (2,4), (3,6), (4,8),
• (1,3) (1,3), (2,6), (3,9), (4,12),
• (2,3) (2,3), (4,6), (6,9), (8,12),
• (2,1) (2,1), (4,2), (6,3), (8,4),
• Looks familiar?
• We have defined what it means for two fractions
  to be the same.
• Distinct Equivalence classes are
• (m,n) where m,nÎ Z and gcd(m,n) 1.

88
8. Summary of Definition Theorems

• Definition Let R be an equivalence relation on
  a set A, and let a Î A. The
  equivalence class of a is defined as follows
• a x Î A x R a
• OR
• x Î a iff x R a
• If a R b then a b
• Either a Ç b f or a b
• Given an Equivalence relation R on A, the
  distinct equivalence classes of R form a
  partition of A.

89
Again the skills

• 2 things you need to know
• Your HEAD
• Logic (which includes proving)
• Definitions (which includes theorems and lemmas)
• Always go back to the definition.
• Do not misinterpret the definition.
• Be very very sensitive to what the definition
  says and what it does not say.
• Be careful that you do not add in your
  pre-conceived ideas into the definition.
• Trust the definition (not your feelings)
• Memorise the definitions.
• Your HEART
• Intuition Draw diagrams if possible, to get a
  good feel of the problem.
• Try various small examples.
• End of Lecture