free video lectures for all courses

1. Video Lectures for MBA
By:
Video.edhole.com

2. Equivalence Relations:
Selected Exercises
Video.edhole.com

3. 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)
Video.edhole.com

4. Copyright © Peter Cappello 4
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?
Video.edhole.com

5. Copyright © Peter Cappello 5
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.
Video.edhole.com

6. Copyright © Peter Cappello 6
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.
Video.edhole.com

7. Copyright © Peter Cappello 7
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.
Video.edhole.com

8. Copyright © Peter Cappello 8
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)
Video.edhole.com

9. Copyright © Peter Cappello 9
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.
Video.edhole.com

10. Copyright © Peter Cappello 10
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.
Video.edhole.com

11. Copyright © Peter Cappello 11
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
Video.edhole.com

12. Copyright © Peter Cappello 2011 12
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)
• 01010101
(answer: all strings that begin with 010)
Video.edhole.com

13. Copyright © Peter Cappello 13
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.
Video.edhole.com

14. Copyright © Peter Cappello 14
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 }.
Video.edhole.com

15. Copyright © Peter Cappello 15
Exercise 40 continued
b) Interpret the equivalence classes of the equivalence
relation R in Exercise 16.
Video.edhole.com

16. Copyright © Peter Cappello 16
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.)
Video.edhole.com

17. Copyright © Peter Cappello 17
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.
Video.edhole.com

18. Copyright © Peter Cappello 18
Exercise 50 continued
It suffices to note that:
Every county is contained within its state:
No county spans 2 states.
Video.edhole.com

19. Copyright © Peter Cappello 19
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?
Video.edhole.com

20. Copyright © Peter Cappello 20
End 8.5
Video.edhole.com

21. Copyright © Peter Cappello 21
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