Discussion

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Discussion 27Closures Equivalence Relations
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Topics

• Reflexive closure
• Symmetric closure
• Transitive closure
• Equivalence relations
• Partitions

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Closure

• Closure means adding something until done.
• Normally adding as little as possible until some
  condition is satisfied
• Least fixed point similarities

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Reflexive Closure

• Reflexive closure of a relation R(r)
• smallest reflexive relation that contains R (i.e.
  fewest pairs added)
• R(r) R ? IA (R is a relation on a set A,
  and IA is the identity relation ? 1s on
  the diagonal and 0s elsewhere.)

R
R(r)
?x (xRx)
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Symmetric Closure

• Symmetric closure of a relation R(s)
• smallest symmetric relation that contains R (i.e.
  fewest pairs added)
• R(s) R ? R (R is R inverse)

R
R
R ? R
?x?y(xRy ? yRx)
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Transitive Closure

• Transitive closure of a relation R(t) R
• smallest transitive relation that contains R
  (i.e. fewest pairs added)
• for each path of length i, there must be a direct
  path. (This follows from x?y, y?z ? x?z since,
  if we also have v?x, we must have v?z, a path of
  length 3.)
• R(t) R ? R2 ? R3 ? ? RA. (No path can be
  longer than A, the number of elements in A.)

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Transitive Closure Example 1
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All paths of length 1
R
2
3
1
All paths of length 2
R2
2
3
1
All paths of length 3
RR2 R3
2
3
R?R2?R3
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Transitive Closure Example 2
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R
All paths of length 1
2
3
1
R2
All paths of length 2
2
3
1
RR2 R3
All paths of length 3
2
3
1
R?R2?R3
Paths of any length
2
3
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Reflexive Transitive Closure

• Reflexive transitive closure of a relation R
• smallest reflexive and transitive relation that
  contains R
• R IA ? R R0 ? R R0 ? R1 ? R2 ? RA
• Example

IA R0
R1 ? R2 ? R3
1
R0 ? R1 ? R2 ? R3
2
3
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Equivalence Relations

• A relation R on a set A is an equivalence
  relation if R is reflexive, symmetric, and
  transitive.
• Equivalence relations are about equivalence
• Examples
• for integers x x reflexive
• x y ? y x symmetric
• xy ? yz ? xz transitive
• for sets A A reflexive
• A B ? B A symmetric
• AB ? BC ? AC transitive
• Let R be has same major as for college students
• xRx ? reflexive same major as self
• xRy ? yRx ? symmetric same major as each
  other
• xRy ? yRz ? xRz ? transitive same as, same as
  ? same as

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Partitions

• A partition of a set S is
• a set of subsets Si1,2,n of S
• such that ?ni1 Si S, Sj ? Sk ? for j ? k.
• Each Si is called a block (also called an
  equivalence class)
• Example
• Suppose we form teams (e.g. for a doubles tennis
  tournament) from the set
• Abe, Kay, Jim, Nan, Pat, Zed
• then teams could be
• Abe, Nan, Kay, Jim, Pat, Zed
• Note on same team as is reflexive, symmetric,
  transitive ? an equivalence relation.
  Equivalence relations and partitions are the same
  thing (two sides of the same coin).

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Partitions (continued)

• Since individual elements can only appear in one
  block (Sj ? Sk ? for j ? k), blocks can be
  represented by any element within the block.
• e.g. Nans Team
• John Stockons 1995 NBA finals team
• Formally, x set of all elements related to x
  and y ? x iff xRy
• e.g. Nan represents Abe, Nan, Nans team
• Abe represents Abe, Nan, Abes team
• John Stockton represents the set of players
  who played in the playoffs for the Jazz in 1995

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Partitions Equivalence Relations
Example

• The mod function partitions the natural numbers
  into equivalence classes.
• 0 mod 3 0 so 0 forms a class 0
• 1 mod 3 1 so 1 forms new class 1
• 2 mod 3 2 so 2 forms new class 2
• 3 mod 3 0 so 3 belongs to 0
• 4 mod 3 1 so 4 belongs to 1
• 5 mod 3 2 so 5 belongs to 2
• 6 mod 3 0 so 6 belongs to 0
• Thus, the mod function partitions the natural
  numbers into equivalence classes.
• 0 0, 3, 6,
• 1 1, 4, 7,
• 2 2, 5, 8,

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Partitions ? Equivalence Relations
Theorem If S1, , Sn is a partition of S, then
RS?S is an equivalence relation, where R is in
same block as. Note to prove that R is an
equivalence relation, we must prove that R is
reflexive, symmetric, and transitive. Proof
Reflexive since every element is in the same
block as itself. Symmetric since if x is in the
same block as y, y is in the same block as x.
Transitive since if x and y are in the same
block and y and z are in the same block, x and z
are in the same block.
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Equivalence Relations ? Partitions
Theorem If RS?S is an equivalence relation and
x y xRy , then x x ? S is a
partition P of S. Note to prove that we have a
partition, we must prove (1) that every element
of S is in a block of P, and (2) that for every
pair of distinct blocks Sj and Sk (j?k) of P, Sj
? Sk ?. Proof (1) Since R is reflexive, xRx,
every element of S is at least in its own block
and thus in some block of P. (2) Suppose Sj ? Sk
? ? for distinct blocks Sj and Sk of P. Then, at
least one element y is in both Sj and Sk. Let Sj
y, x1, xn and Sk y, z1, zm, then
yRxi, i1, 2, , n, and yRzp, p1, 2, , m.
Since R is symmetric, xiRy, and since R is
transitive xiRzp. But now, since the elements of
Sj are R-related to the elements of Sk, x1, ,
xn, y, z1, , zm are together in a block of P and
thus Si and Sk are not distinct blocks of P.