• 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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2

• 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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3

A partition of set S is a set of nonempty subsets ,
S
1
, S
2
, . . ., S n
, of S such that:
1.
 i
 j ( i ≠ j 
S i
∩ S j
= Ø ).
2. S = S
1
U S
2
U . . . U S n
.
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4

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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5

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)
1. c ~ d ( c, d

[a] )
2. d ~ b ( d

[b] )
3. c ~ b ( c ~ d
 d ~ b

E is transitive )
2.
The union of the equivalence classes is S .
Students: Show this use pair proving in class.
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6

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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7

• 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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8

• That is, x R y 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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9

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., 101 11 ~ 101 000).
a) 010 b) 1011 c) 11111 d) 01010101
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10

• 010
(answer: all strings that begin with 010)
• 1011
(answer: all strings that begin with 101)
• 11111
(answer: all strings that begin with 111)
• 01010101
(answer: all strings that begin with 010)
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11

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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( a, b ) ~ ( c, d )
 ad = bc
 a/b = c/d
[ ( 1, 2 ) ] = { ( c, d ) | ( 1, 2 ) ~ ( c, d ) }
= { ( c, d ) | 1 d = 2 c
 c/d = ½ }.
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13

b) Interpret the equivalence classes of the equivalence relation R in Exercise 16.
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14

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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15

• 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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16

It suffices to note that:
Every county is contained within its state:
No county spans 2 states.
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17

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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18

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19

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 ).
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