Equivalence Relation

1
Equivalence Relation

• Section 4.5

2
Equivalence Relation

• A relation R on a set A is called an equivalence
  relation if it is reflexive, symmetric, and
  transitive
• Let Aa,b,c,d and let R(a,a), (a,b),(b,a),
  (b,b), (c,d),(d,c), (c,c),(d,d), is R an
  equivalence relation?
• Reflexive? (a,a),(b,b),(c,c),(d,d)
• Symmetric? (a,b),(b,a) (c,d),(d,c)
• Transitive
• Compute (MR)2?MR
• Ex.3.

a b c d
a 1 1 0 0
b 1 1 0 0
c 0 0 1 1
d 0 0 1 1
3
Congruent

• AZ and let R(a,b) ? A X A a and b yield the
  same remainder when divided by 2
• 2 is the modulus
• a b (mod 2)
• a is congruent to b mod 2
• Congruent mod 2 is an equivalence relation
• a a (mod 2)?
• b a (mod 2)?
• a b (mod 2), and b c (mod 2), then a c (mod
  2)?
• Ex. 5

4
Equivalence relation determined by P

• Theorem 1
• Let P be a partition of a set A (sets in P is
  called block of P). Define the relation
• R a R b iff a and b are member of the same block
• Then R is an equivalence relation of A
• Proof
• a R a, a is in the same block of itself
• a R b the a R b
• a R b and b R c, then a R c
• R is called the Equivalence relation determined
  by P
• How to find the equivalence relation determined
  by P
• Each element in a block is related to every other
  element in the same block
• Ex. 6

5
Equivalence class

• If R is an equivalence relation on A, then the
  sets R(a) are transitionally called equivalence
  classes of R.
• a
• A1,2,3,4 and R(1,1), (1,2), (2,1), (2,2),
  (3,4), (4,3), (3,3),(4,4)
• R(1)1,2
• R(2)1,2
• R(3)3,4
• R(4)3,4
• Quotient sets of A
• Denote A/R
• 1,2,3,4
• Is partitions of A