1. Equivalence Relations: Selected Exercises

2. Copyright © Peter Cappello 2011
Equivalence Relation
Let E be a relation on set A.
E is an equivalence relation 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 } and is 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:
i j ( i ≠ j  Si ∩ Sj = Ø ).
S = S1 U S2 U U Sn.
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5. 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.
[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 ~ a ( c  [a] )
a ~ d ( d  [a] )
c ~ d ( c ~ a  a ~ d  E is transitive )
d ~ b ( d  [b] )
c ~ 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.
Let P be an arbitrary partition of S.
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., ~ ).
010
1011
11111
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Exercise 30 Answer
010
(answer: all strings that begin with 010)
1011
(answer: all strings that begin with 101)
11111
(answer: all strings that begin with 111)
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Exercise 40
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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Exercise 40 continued
b) Interpret the equivalence classes of the equivalence relation R in Exercise 16.
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Exercise 40 continued
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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Exercise 50 continued
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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20. Copyright © Peter Cappello 201110
Suppose A   & R is an equivalence relation on A.
Show f X f: A  X such that a ~ b  f( a ) = f( b ).
Proof.
Let f : A  X, where
X = { [a] | [a] is an equivalence class of R }
a f (a ) = [a].
Then, a b a ~ b  f( a ) = [a] = [b] = f( b ).
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Use Thm. 2 to form an equivalence relation R from a partition P of a set A.
What partition results from applying Thm. 2 to R? (i.e, the partition defined by R’s equivalence classes)?
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P = {S1 , S2 , S3 , S4 , S5 , S6} S2 S3 S1 S6 S4 S5
P =
A = S1  S2  S3  S4  S5  S6
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66 continued
Let P = { S1, S2, … , Sn }.
R = { ( a, b ) | S  P a,b  S }.
Let P’ be the partition defined by R’s equivalence classes.
By defn. a  A [a] = { b | aRb }.
But aRb  a,b  S, for some S  P.
Therefore, P’ = P.
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