1. Equivalence Relations
Lecture 45
Section 10.3
Fri, Apr 8, 2005

2. Equivalence Relations
An equivalence relation on a set A is a relation on A that is reflexive, symmetric, and transitive.
We often use the symbol ~ as a generic symbol for an equivalence relation.

3. Examples of Equivalence Relations
Which of the following are equivalence relations?
a  b, on Z+.
gcd(a, b) > 1, on Z+.
A  B, on (U).
p  q, on a set of statements.
p  q, on a set of statements.
a  b (mod 10), on Z.

4. Examples of Equivalence Relations
Which of the following are equivalence relations?
p  q = p, on a set of statements.
gcd(a, b) = 1, on Z+.
gcd(a, b) = a, on Z+.
A  B = , on (U).
A = B, on (U).

5. Examples of Equivalence Relations
Which of the following are equivalence relations?
R  R, on R.
, on R.

6. Equivalence Classes
Let ~ be an equivalence relation on a set A and let a  A.
The equivalence class of a is
[a] = {x  A  x ~ a}.

7. Examples: Equivalence Classes
Describe the equivalence classes of each of the following equivalence relations.
a  b (mod 10), on Z.
A = B, on (U).
p  q, on a set of statements.
R  R, on R.

8. Equivalence Classes and Partitions
Theorem: Let ~ be an equivalence relation on a set A. The equivalence classes of ~ form a partition of A.
Proof:
We must show that
The equivalence classes are pairwise disjoint,
The union of the equivalence classes equals A.

9. Equivalence Classes and Partitions
Proof that the equivalence classes are pairwise disjoint.
Let [a] and [b] be two distinct equivalence classes.
Suppose [a]  [b]  .
Let x  [a]  [b].
Then x ~ a and x ~ b.
Therefore, a ~ x and x ~ b.

10. Equivalence Classes and Partitions
By transitivity, a ~ b.
Now let y  [a].
Then y ~ a.
By transitivity, y ~ b.
So y  [b].
Therefore, [a]  [b].
By a similar argument, [b]  [a].

11. Equivalence Classes and Partitions
Thus, [a] = [b], which is a contradiction
Therefore, [a]  [b] = .
Thus, the equivalence classes are pairwise disjoint.

12. Equivalence Classes and Partitions
Proof that the union of the equivalence classes is A.
Let a  A.
Then a  [a] since a ~ a.
Therefore, a is in the union of the equivalence classes.
So, A is a subset of the union of the equivalence classes.

13. Equivalence Classes and Partitions
On the other hand, every equivalence class is a subset of A.
Therefore, the union of the equivalence classes is a subset of A.
Therefore, the union of the equivalence classes equals A.
Therefore, the equivalence classes form a partition of A.

14. Example Let F be the set of all functions f : R  R.
For f, g  F, define f ~ g to mean that f is (g).

15. Example Theorem: ~ is an equivalence relation on F. Proof: Reflexivity
Obviously, f ~ f for all f  F.

16. M1g(x)  f(x)  M2g(x),
Example
Symmetry
Suppose that f ~ g for some f, g  F.
Then f(x) is (g(x)).
There exist positive constants M1, M2, and x0 such that
M1g(x)  f(x)  M2g(x),
for all x > x0.

17. (1/M2)f(x)  g(x)  (1/M1)f(x),
Example
It follows that
(1/M2)f(x)  g(x)  (1/M1)f(x),
for all x > x0.
Therefore, g(x) is (f(x)).

18. Example Transitivity Let f, g, h  F and suppose that f ~ g and g ~ h.
Then there exist constants M1 and x1 and M2 and x2 such that
f(x) M1g(x)
for all x  x1 and

19. Example g(x) M2h(x) for all x  x2. Let x0 = max(x1, x2).
Then for all x  x0,
f(x) M1g(x)
 M1  M2h(x)
Therefore, f(x) is O(h(x)).

20. Example Similarly, we can show that h(x) is O(f(x)).
Therefore, f(x) is (h(x)).
Therefore, f ~ h.
Therefore, ~ is an equivalence relation on F.

21. Example
The equivalence class of f is the set [f] of all functions with the same growth rate as f.
The most important equivalence classes are
[xa], a  R, a > 0.
[bx], b  R, b > 1.
[xa logb x], a  R, a > 0, b > 1.

22. Example Furthermore, However, [xa]  [xb] if a  b.
[ax]  [bx] if a  b.
However,
[loga x] = [logb x] for all a, b > 1.

23. The Equivalence Relation Induced by a Partition
Let A be a set and let {Ai}iI be a partition of A.
Define a relation ~ on A as
x ~ y  x, y  Ai for some i  I.

24. The Equivalence Relation Induced by a Partition
Theorem: The relation ~ defined above is an equivalence relation on A.

25. The Equivalence Relation Induced by a Partition
Proof:
We must prove that ~ is reflexive, symmetric, and transitive.
Proof that ~ is reflexive.
Let a  A.
Then a is in Ai for some i  I.
So a ~ a.

26. The Equivalence Relation Induced by a Partition
Proof that ~ is symmetric.
Let a, b  A and suppose that a ~ b.
Then a, b  Ai for some i  I.
So b, a  Ai for some i  I.
Therefore b ~ a.

27. The Equivalence Relation Induced by a Partition
Proof that ~ is transitive.
Let a, b, c  A and suppose a ~ b and b ~ c.
Then a, b  Ai for some i  I and b, c  Aj for some j  I.
That means that b  Ai  Aj.
This is possible only if Ai = Aj.
Therefore, a, c  Ai.
So, a ~ c.

28. Example Consider the set P of all computer programs.
Partition P into subsets by putting in the same subset any two programs that always produce identical output for the same input.

29. Example This partition determines an equivalence relation  on P.
Let p1 and p2 be two computer programs.
Then p1  p2 if p1 and p2 always produce identical output for the same input.

30. Example Let A be the set of all people on Earth.
Let R be the relation defined by x R y if x and y have ever shaken hands.
Is R reflexive? Symmetric? Transitive?
Let R* be the reflexive-transitive closure of R.
Is R* an equivalence relation?
What are the equivalence classes of R*?