Orders and Equivalences - Washington University in...

OutlineEquivalence Relations

PartitionsCongruences

Orders

Orders and Equivalences

Gazihan Alankus(Based on original slides by Brahim Hnich et al.)

August 9, 2012

Gazihan Alankus (Based on original slides by Brahim Hnich et al.)Orders and Equivalences



Orders

Equivalence Relations

Partitions

Congruences

Orders




Orders

Equivalence Relations

Definition (Equivalence Relation)

A binary relation R on set A is said to be an equivalence relation ifit is reflexive, symmetric, and transitive.

I Let R ⊆ People × People. Pair (a, b) ∈ R if and only if a andb are of the same age.

I Let R ⊆ Animals × Animals. Pair (a, b) ∈ R if and only if aand b belong to same species.

I Let R ⊆ Students × Students. Pair (a, b) ∈ R if and only if aand b belong to same gender.




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Equivalence Classes

I Let R be an equivalence relation on set A.

I Take a ∈ A. The set C (a) = {b|(a, b) ∈ R} is called theequivalence class of a.

I For example, if ‘Ali’ is 34 years old, then C (‘Ali’) is the set ofall 34 year old people.




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Equivalence (cntd)

LemmaFor any a ∈ A, the class C (a) 6= ∅

Proof.R is reflexive, therefore, (a, a) ∈ R. Hence a ∈ C (a). Thus,C (a) 6= ∅.




Orders

Equivalence (cntd)

LemmaIf C (a) 6= C (b) then C (a) ∩ C (b) = ∅

Proof.Proof by contraposition. Suppose c ∈ C (a) ∩ C (b). Hence,(a, c), (b, c) ∈ R. By symmetricity (c , b) ∈ R. Then, bytransitivity, (a, b) ∈ R. Take x ∈ C (b). We have (b, x) ∈ R. Bytransitivity, (a, x) ∈ R. Hence, x ∈ C (a). Thus C (b) ⊆ C (a).The case of C (a) ⊆ C (b) is similar.




Orders

Equivalence (cntd)

LemmaA =

⋃a∈A C (a)

Proof.Obvious because a ∈ C (a) for all a ∈ A




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Partitions

The equivalence classes divide set A into disjoint subsets.

Definition (Partition)

A collection of subsets M1, . . . ,Mn of a set A is called a partition ifthe following conditions hold.(1) Every Mi 6= ∅(2) If Mi 6= Mj then Mi ∩Mj = ∅(3) A =

⋃ni=1Mi




Orders

Partitions and Equivalence Relations

Lemma shows that the equivalence classes constitute a partition ofthe set. Actually, a stronger statement is true.

TheoremLet A be a set.

I If R is an equivalence relation on A, then its equivalenceclasses form a partition on A.

I If M1, . . . ,Mn is a partition of set A, then the relation Rdefined as follows: (a, b) ∈ R if and only if a, b ∈ Mi for somei , is an equivalence relation on A.

Proof.homework




Orders

Congruences

Definition (Congruency)

Let k be an integer. Integers a, b are congruent modulo k, if theirreminders are equal when divided by k ,or, equivalently, if k divides a− b.

Congruency of a and b modulo k is denoted by a ≡ b(mod k)

Example:−4 ≡ −1(mod 3)




Orders

Congruences (cntd)

I The relation ≡ (mod k), ’to be congruent modulo k ’ isI reflexive, because k divides a− a = 0I symmetric, because if k divides a− b then it also divides b − aI transitive, because if k divides a− b and b − c , then it also

divides a− c = (a− b) + (b − c)

I ≡ (mod k) is an equivalence relation with equivalence classesC (c) = {a|∃b (a = bk + c)}

I Arithmetic on these classes are called modular arithmetic




Orders

Orders

I A relation R on set A is called a (partial) order if it isreflexive, transitive, and anti-symmetric.

I Example: a ≤ b on the set of real numbers

I Example: (a, b) ∈ Div if and only if a divides b




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Diagrams of Partial Orders

Due to anti-symmetricity, all the elements of Aare ranked with respect to the order R. That isb is ranked higher than a if (a, b) ∈ R.




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Due to transitivity, we do not need to knowall pairs (a, b) from relation, but only those inwhich b is just higher than a.

Connect every element only with elements thatare just higher, so avoid triangles




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Example: Show diagram for relation of divisibility on {1, 2, . . . , 12}




Orders


I Elements a, b are said to be comparable if (a, b) ∈ R or(b, a) ∈ R; Otherwise they are called incomparable

I Element a is minimal if ∀(b ∈ A) (((b, a) ∈ R)→ (a = b))

I Element a is maximal if ∀(b ∈ A) (((a, b) ∈ R)→ (a = b))

I Element a is the least if ∀(b ∈ A) ((a, b) ∈ R)

I Element a is the greatest if ∀(b ∈ A) ((b, a) ∈ R)




Orders

Total Orders

I A partial order is said to be total if every two elements arecomparable.

I Sets N , Z, Q, R are all totally ordered with respect to ≤I The diagram for a total order is a chain


Orders and Equivalences - Washington University in...

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