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.
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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.
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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.
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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.
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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!

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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.