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Equivalence Relations: Selected Exercises

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Copyright © Peter Cappello 3

Equivalence Relation

• Let E be a relation on set A.

• E is an equivalence relation if & only if it is:

– Reflexive

– Symmetric

– Transitive.

• Examples

– a E b when a mod 5 = b mod 5. (Over N)

(i.e., a ≡ b mod 5 )

– a E b when a is a sibling of b. (Over humans)

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Equivalence Class

• Let E be an equivalence relation on A.

• We denote aEb as a ~ b. (sometimes, it is denoted a ≡ b )

• The equivalence class of a is { b | a ~ b }, denoted [a].

• What are the equivalence classes of the example equivalence

relations?

• For these examples:

– Do distinct equivalence classes have a non-empty intersection?

– Does the union of all equivalence classes equal the underlying set?

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Partition

A partition of set S is a set of nonempty subsets, S1,

S2, . . ., Sn, of S such that:

1. ∀i ∀j ( i ≠ j → Si ∩ Sj = Ø ).

2. S = S1 U S2 U . . . U Sn.

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Equivalence Relations & Partitions

Let E be an equivalence relation on S.

• Thm. E’s equivalence classes partition S.

• Thm. For any partition P of S, there is an equivalence

relation on S whose equivalence classes form partition P.

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E’s equivalence classes partition S.

1. [a] ≠ [b] → [a] ∩ [b] = Ø.

Proof by contradiction:

Assume [a] ≠ [b] ∧ [a] ∩ [b] ≠ Ø: (Draw a Venn diagram)

Without loss of generality, let c ∈ [a] - [b]. Let d ∈ [a] ∩ [b].

We show that c ∈ [b] (which contradicts our assumption above)

– c ~ d ( c, d ∈ [a] )

– d ~ b ( d ∈ [b] )

– c ~ b ( c ~ d ∧ d ~ b ∧ E is transitive )

• The union of the equivalence classes is S.

Students: Show this use pair proving in class.

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For any partition P of S, there is an equivalence relation

whose equivalence classes form the partition P.

Prove in class.

1. Let P be an arbitrary partition of S.

2. We define an equivalence relation whose

equivalence classes form partition P.

(Students: Show this (use pair proving) in class)

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Exercise 20

• Let P be the set of people who visited web page W.

• Let R be a relation on P: xRy ↔ x & y visit the same

sequence of web pages since visiting W until they exit the

browser.

• Is R an equivalence relation?

• Let s( p ) be the sequence of web pages p visits since

visiting W until p exits the browser.

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Exercise 20 continued

• That is, xRy means s( x ) = s( y ).

∀ ∀x xRx: R is reflexive.

Since ∀x s( x ) = s( x ).

∀ ∀x ∀y ( xRy → yRx ): R is symmetric.

Since s( x ) = s( y ) → s (y ) = s( x ).

∀ ∀x ∀y ∀z ( ( xRy ∧ yRz ) → xRz ): R is transitive.

Since ( s( x ) = s( y ) ∧ s( y ) = s( z ) ) → s( x ) = s( z ).

• Therefore, R is an equivalence relation.

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Exercise 30

What are the equivalence classes of the bit strings for the equivalence relation of Exercise 11?

Ex. 11: Let S = { x | x is a bit string of ≥ 3 bits. }

Define xRy such that x agrees with y on the left 3 bits (e.g., 10111 ~ 101000).

a) 010

b) 1011

c) 11111

d) 01010101

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Copyright © Peter Cappello 2011 12

Exercise 30 Answer

• 010

(answer: all strings that begin with 010)

• 1011


• 11111


• 01010101


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Exercise 40

a) What is the equivalence class of (1, 2) with respect

to the equivalence relation given in Exercise 16?

Exercise. 16:

Ordered pairs of positive integers such that

( a, b ) ~ ( c, d ) ↔ ad = bc.

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Exercise 40 a) Answer

( a, b ) ~ ( c, d ) ↔ ad = bc ↔ a/b = c/d

[ ( 1, 2 ) ] = { ( c, d ) | ( 1, 2 ) ~ ( c, d ) }

= { ( c, d ) | 1d = 2c ↔ c/d = ½ }.

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b) Interpret the equivalence classes of the equivalence

relation R in Exercise 16.

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b) Interpret the equivalence classes of the equivalence

relation R in Exercise 16.

Answer

Each equivalence class contains all (p, q), which, as

fractions, have the same value (i.e., the same

element of Q+).

(The fact that 3/7 = 15/35 confuses some small children.)

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Exercise 50

• A partition P’ is a refinement of partition P when

∀x ∈ P’ ∃y ∈ P x ⊆ y. (Illustrate.)

• Let partition P consist of sets of

people living in the same US state.

• Let partition P’ consist of sets of

people living in the same county of a state.

• Show that P’ is a refinement of P.

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It suffices to note that:

Every county is contained within its state:

No county spans 2 states.

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Exercise 62

Determine the number of equivalent relations on a set

with 4 elements by listing them.

How would you represent the equivalence relations

that you list?

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End 8.5

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10

Suppose A ≠ ∅ & R is an equivalence relation on A.

Show ∃f ∃X f: A → X such that a ~ b ↔ f( a ) =

f( b ).

Proof.

1. Let f : A → X, where

1. X = { [a] | [a] is an equivalence class of R }

2. ∀a f (a ) = [a].

2. Then, ∀a ∀b a ~ b ↔ f( a ) = [a] = [b] = f( b ).Video.edhole.com

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