Equivalence Relations

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Equivalence Relations
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Equivalence Relations

• A relation R on A?A is an equivalence relation
  when R satisfies 3 conditions
• ?x ? A, xRx (reflexive).
• ?x, y ? A, xRy ? yRx (symmetric).
• ?x, y, z ? A, (xRy ? yRz) ? xRz (transitive).
• How are the properties of an equivalence relation
  reflected in its graph representation?

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Examples?

• Let P be the set of all human beings R ? P?P.
• Is R an equivalence relation if
• aRb when a is the brother of b?
• aRb when a is in the same family as b?
• R ? N? N.
• xRy when x has the same remainder as y when they
  are divided by 5?

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Examples?

• Let R2 R?R be the set of points in the plane.
• Is E ? R2 ? R2 when
• E1 ((x1,y1),(x2,y2)) (x1,y1) (x2,y2) are
  on the same horizontal line?
• (E1 partitions the plane into horizontal
  lines.)
• E2 ((x1,y1),(x2,y2)) (x1,y1) (x2,y2) are
  equidistant from the origin?
• (E2 partitions the plane into concentric
  circles.)

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Partitions

• Let S be a set. A partition, ?(S), of S is a set
  of nonempty subsets of S such that
• ?Si ? ?(S) Si S (the parts cover S)
• ?Si?Sj ? ?(S), Si?Sj ? (the parts are disjoint)
• Example (check 2 conditions of partition)
• The same remainder when divided by 5 partitions
  N into 5 parts.
• E1 partitions the plane into horizontal lines.
• E2 partitions the plane into concentric circles.

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Partitions as Equivalence Relations

• Let E ? S?S be an equivalence relation, and a ?S.
• The equivalence class determined by a is
• a b ?S aEb the set of all elements of S
  equivalent to a.
• Let P be the set of equivalence classes under E.

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aEb ? a b

• I.e., any member of a can name the class.
• Assume a?b Without loss of generality, ?c
  ?b and c?a (draw a Venn diagram)
• aEb, (given)
• bEc, (by assumption)
• aEc, (E is transitive)
• c ?a (definition of equivalence class).
• Therefore, a?b is false.

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Equivalence classes partition S

• To prove this, we must show that
• (i) the union of all equivalence classes equals
  S
• (ii) if a is not equivalent to b, then a ?b
  ?.
• (i) Since E is reflexive, ?a ?S, there is some
  equivalence class that contains a a.
• Therefore, ?a ? S a S.

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Equivalence classes partition S ...

• (ii) To show For a?b, a ?b ?
• Assume not ? c ? a ?b.
• c ? a ? aEc which implies a c
• c ? b ? bEc which implies b c
• Therefore, a b, a contradiction.
• Therefore, a ?b ?.
• The set of equivalence classes partitions S.

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A partition defines an equivalence relation

• Let ?(S) be a partition of S
• ?Si ? ?(S) Si S (the parts cover S)
• ?Si?Sj ? ?(S), Si?Sj ? (the parts are disjoint)
• Define ?E (a,b) a, b ?Si ? ?(S).
• Illustrate on blackboard.
• Claim ?E is an equivalence relation
• ?E is reflexive, symmetric, transitive.

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?E is an equivalence relation

• (i) ?x ? S, x ?E x (reflexive)
• Since ?(S) is a partition, every x is in some
  part.
• Every element x of S is in the same part as
  itself x ?E x.
• (ii)?x, y ? S, x ?E y ? y ?E x (symmetric).
• If x is in the same part as y, then y is in the
  same part as x.

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?E is an equivalence relation ...

• (iii) ?x, y, z ? S, (x ?E y ? y ?E z) ? x ?E z
  (transitive)
• If x is in the same part as y and y is in the
  same part as z, then x is in the same part as z.
• ?E is an equivalence relation

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Equivalence relations summary

• Partitioning a set S is the same thing as
  defining an equivalence relation over S.
• If E is an equivalence relation of S, the
  associated partition is called the quotient set
  of S relative to E and is denoted S/E.