Equivalence Relations

1
Section 10.3

• Equivalence Relations

2
Basic Idea

• We want to group together elements that can be
  considered the same somehow.
• Youve done this before with fractions you know
  that 1/3 2/6 3/9 27/81, etc.

3
The Relation Induced by a Partition

• Given a partition of a set A, the binary relation
  induced by the partition, R, is defined on A as
  follows
• For all x, y ? A, x R y ? there is a subset Ai of
  the partition such that both x and y are in Ai

4
Example Let A 0, 1, 2, 3, 4, and consider
the following partition of A 0, 2, 1, 3, 4

• Note that 0 R 0, 0 R 2, 2 R 0, and 2 R 2 because
  0 and 2 are in the same subset. The
  relationships for other subsets are similar.
• R (0, 0), (0, 2), (1, 1), (2, 0), (2, 2), (3,
  3), (3, 4), (4, 3), (4, 4)

5
Theorem Let A be a set with a partition and let
R be the relation induced by the partition. Then
R is reflexive, symmetric, and transitive.

• Proof Suppose A is a set with a partition A1,
  A2, , An that has only a finite number of sets.
  (The proof for a partition with an infinite
  number of sets is really the same, with different
  notation.) Then Ai ? Aj ? whenever i ? j and
  A1 ? A2 ? ? An A. The relation R induced by
  this partition is defined as follows for all x,
  y ? A,
• x R y ? there is a set Ai of the partition such
  that
• x, y ? Ai.

6
Theorem Let A be a set with a partition and let
R be the relation induced by the partition. Then
R is reflexive, symmetric, and transitive.

• R is reflexive Suppose x ? A. Since A1, A2, ,
  An is a partition of A, we know that x is a
  member of one of these Ai for some index i.
  Moreover, x does not change subsets. Therefore,
  x R x by the definition of R.

7
Theorem Let A be a set with a partition and let
R be the relation induced by the partition. Then
R is reflexive, symmetric, and transitive.

• R is symmetric Suppose x and y are elements of A
  such that x R y. Then x and y are in the same
  subset Ai of our partition of A. But this means
  that y R x, because the statement y and x are in
  subset Ai of the partition is true.

8
Theorem Let A be a set with a partition and let
R be the relation induced by the partition. Then
R is reflexive, symmetric, and transitive.

• R is transitive Suppose that x, y, and z are in
  A and that x R y and y R z. By the definition of
  R this means there are subsets Ai and Aj of the
  partition such that x and y are in Ai and y and z
  are in Aj. Suppose i ? j. Then y is in two
  different subset in the partition, which is
  impossible by the definition of a partition.
  Therefore, x, y, and z must be in the same set Ai
  or the partition, which implies that x R z.

9
Equivalence Relation

• Let A be a nonempty set and R a binary relation
  on A. R is an equivalence relation if, and only
  if, R is reflexive, symmetric, and transitive.
• Soany relation induced by a partition is an
  equivalence relation.

10
Notation

• Let m and n be integers and let d be a positive
  integer. The notation
• m ? n (mod d)
• is read m is congruent to n modulo d and means
  that
• d m-n.
• This is an equivalence relation.

11
Evaluating Congruences

• 13 ? 8 (mod 5) because 13 - 8 5, which is
  divisible by 5.
• 13 ? 7 (mod 5) because 13 7 6, which is not
  divisible by 5.
• 13 ? -12 (mod 5) because 13 (-12) 25, which
  is divisible by 5.

12
A Binary Relation on a Set of Identifiers

• Let L be the set of all allowable identifiers in
  a certain computer language, and define a
  relation R on L as follows for all strings s and
  t in L,
• s R t ? the first eight characters of s and t are
  the same
• Prove that R is an equivalence relation on L.

13
R is reflexive

• Let s ? L. Clearly s has the same first eight
  characters as itself, so by the definition of R,
  s R s.

14
R is symmetric

• Let s and t be in L and suppose that s R t. By
  the definition of R, s and t must have the same
  first eight characters. But then the first eight
  characters of t are the same as the first eight
  characters of s, which means that t R s.

15
R is transitive

• Let s, t, and u be three elements of L such that
  s R t and t R u. Then the first eight characters
  of s are the same as the first eight characters
  of t and the first eight characters of t are the
  same as the first eight characters of u. But
  this means the first eight characters of s are
  the same as the first eight characters of u, so s
  R u.

16
Friday

• A little bit on equivalence classes.
• How this course ties in to Theory I.