Equivalence relations

1
Equivalence relations

• Binary relations
• Let S1 and S2 be two sets, and R be a (binary
  relation) from S1 to S2
• Not every x in S1 and y in S2 have such relation
• If R holds for a in S1 and b in S2, denote as aRb
  or R(a, b)
• Examples
• Spouse relation from set Men to set Women
• S1 and S2 can be the same set
• Examples
• Parent relation on set Human
• gt (greater than) relation on set Z (all integers)

2
Equivalence relations (cont)

• Properties of binary relations
• Let R be a binary relation on set S
• R is reflexive if aRa for all a in S
• Ex relation, gt relation
• R is symmetric aRb iff bRa
• Ex relation, spouse relation
• R is transitive if aRb and bRc, then aRc
• Ex relation, gt relation, ancestor relation
• R is an equivalence relation if it is reflexive,
  symmetric, and transitive.
• Ex. relation, relative relation among humans
• Counter ex gt relation, spouse relation
• Use to denote an abstract generic equivalence
  relation
• ab

3
Equivalence relations (cont)

• Equivalence classes
• Let be a equivalence relation defined on set S
• S can be partitioned into disjoint subsets such
  that
• If a b, then a and b are in one subset
• If a and b are in two different subsets, then a
  b does not hold
• Each of such subsets is called an equivalence
  class (with respect to relation ), denoted C1,
  C2, ...
• All elements in an equivalence class relate to
  each other by
• No elements in different equivalence classes
  relate to each other by
• Equivalence classes can be represented as
  disjoint sets