Lecture 4.4: Equivalence Classes and Partially Ordered Sets

CS 250, Discrete Structures, Fall 2012

Nitesh Saxena

Adopted from previous lectures by Cinda Heeren

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HW4 HW4 has been posted

due at 11am on Nov 15 (Thursday) Covers the chapter on Relations (lectures

4.*) Has a 10 pointer bonus problem

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Outline

Equivalence Classes Partially Ordered Sets (POSets) Hasse Diagrams

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Equivalence ClassesLemma: Let R be an equivalence relation on

S. Then1. If aRb, then [a]R = [b]R

2. If not aRb, then [a]R [b]R =

Proof:1. Suppose aRb, and consider x S.

x [a]R aRx

xRa

xRb

bRx

x [b]R

Defn of [a]R

symmetry

transitivity

symmetry

Defn of [b]R

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Equivalence ClassesLemma: Let R be an equivalence relation on S.

Then1. If aRb, then [a]R = [b]R

2. If not aRb, then [a]R [b]R =

Proof:2. Suppose to the contrary that x [a]R [b]R.

x [a]R x [b]R aRx and bRx

aRb, contradicting “not aRb”

aRx and xRb

Thus, [a]R and [b]R are either identical or disjoint.

S [a]RaS

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Equivalence ClassesSo S is the union of disjoint equivalence

classes of R.

A partition of a set S is a (perhaps infinite) collection of sets {Ai} with

Each Ai non-empty

Each Ai S

For all i, j, Ai Aj = S = Ai

Each Ai is called a block of the

partition.

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Equivalence ClassesGive a partition of the reals into 2 blocks(numbers <= 0) and (numbers > 0)

Give a partition of the integers into 4 blocksnumbers modulo 4

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Equivalence ClassesTheorem: if R is a _____ S, then {[a]R : a S} is a _____

S.A. Partition of, equivalence relation

onB. Subset of, equivalence class ofC. Relation on, partition ofD. Equivalence relation on, partition

ofE. I have no clue.

Theorem: if R is an equivalence relation on S, then {[a]R : a S} is a partition of S.

Proof: we need to show that an equivalence relation R satisfies the definition of a partition. Follows from previous arguments and definition of equivalence classes.

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Equivalence ClassesTheorem: if {Ai} is any partition of S, then there exists

an equivalence relation R, whose equivalence classes are exactly the blocks Ai.

Proof: If {Ai} partitions S then define relation R on S to be R = {(a,b) : i, a Ai and b Ai}

Next show that R is an equivalence relation.Reflexive and symmetric. Transitive?

Suppose aRb and bRc. Then a and b are in Ai, and b and c are in Aj.

But b Ai Aj, so Ai = Aj.

So, a, b, c Ai, thus aRc.

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Example: Partition Equivalence Relation

Give an equivalence relation for the partition:{1,2}, {3, 4, 5}, {6} for the set {1,2,3,4,5,6}

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Partially Ordered Sets (POSets)

Let R be a relation then R is a Partially Ordered Set (POSet) if it isReflexive - aRa, aTransitive - aRb bRc aRc, a,b,cAntisymmetric - aRb bRa a=b, a,b

Ex. (R,), the relation “” on the real numbers, is a partial order.

How do you check?

a a for any real Reflexive?

Transitive?

Antisymmetric?

If a b, b c then a c

If a b, b a then a = b

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Partially Ordered Sets (POSets)Let (S, R ) be a PO. If a R b, or b R a, then a and b are

comparable. Otherwise, they are incomparable.

A total order is a partial order where every pair of elements is comparable.

Ex. (R, ), is a total order, because for every pair (a,b) in RxR, either a b, or b a.

Ex. (people in a queue, “behind or same place”) is a total order

Ex. (employees, “supervisor”) is not a total order

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Partially Ordered Sets (POSets)Ex. (Z+, | ), the relation “divides” on positive integers.

Yes, x|x since x=1x (k=1)Reflexive?

Transitive?

Antisymmetric?

a|b means b=ak, b|c means c=bj. Does c=am for some m?

c = bj = akj (m=kj)

a|b means b=ak, b|a means a=bj. But b = bjk (subst) only if

jk=1.jk=1 means j=k=1,

and we have b=a1, or b=a

Yes, or No?

A total order?

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Partially Ordered Sets (POSets)

Ex. (Z, | ), the relation “divides” on integers.

Yes, x|x since x=1x (k=1)Reflexive?

Transitive?

Antisymmetric?

a|b means b=ak, b|c means c=bj. Does c=am for some m?

c = bj = akj (m=kj)

3|-3, and -3|3, but 3 -3.

Not a poset.

Yes, or No?

A total order?

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Partially Ordered Sets (POSets)Ex. (2S, ), the relation “subset” on set of all subsets of S.

Yes, A A, A 2SReflexive?

Transitive?

Antisymmetric?

A B, B C. Does that mean A C?

A B means x A x B

A B, B A A=B

A poset.

B C means x B x C

Now take an x, and suppose it’s in A. Must it also be in

C?

A total order?

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Partially Ordered Sets (POSets)

When we don’t have a special relation definition in mind, we use the symbol “” to denote a partial order on a set.

When we know we’re talking about a partial order, we’ll write “a b” instead of “aRb” when discussing members of the relation.

We will also write “a < b” if a b and a b.

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Partially Ordered Sets (POSets)Ex. A common partial order on bit strings

of length n, {0,1}n, is defined as:a1a2…an b1b2…bn

If and only if ai bi, i.0110 and 1000 are “incomparable” … We can’t tell

which is “bigger.”

A. 0110 1000

B. 0110 0000

C. 0110 1110

D. 0110 1011

A total order?

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Hasse DiagramsHasse diagrams are a special kind of

graphs used to describe posets.

Ex. In poset ({1,2,3,4}, ), we can draw the following picture to describe the relation.

1. Draw edge (a,b) if a b

2. Don’t draw up arrows

3. Don’t draw self loops

4. Don’t draw transitive edges

4

3

2

1

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Today’s Reading Rosen 9.5 and 9.6

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