Equivalence Relations Selected Exercises Equivalence Relation Let E

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Equivalence Relations: Selected Exercises

Equivalence Relations: Selected Exercises

Equivalence Relation • Let E be a relation on set A. • E is

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) Copyright © Peter Cappello 2

Equivalence Class • Let E be an equivalence relation on A. • We denote

Equivalence Class • Let E be an equivalence relation on A. • We denote a. Eb 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? Copyright © Peter Cappello 3

Partition A partition of set S is a set of nonempty subsets, S 1,

Partition A partition of set S is a set of nonempty subsets, S 1, S 2, . . . , Sn, of S such that: 1. i j ( i ≠ j Si ∩ Sj = Ø ). 2. S = S 1 U S 2 U. . . U Sn. Copyright © Peter Cappello 4

Equivalence Relations & Partitions Let E be an equivalence relation on S. • Thm.

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. Copyright © Peter Cappello 5

E’s equivalence classes partition S. 1. [a] ≠ [b] [a] ∩ [b] = Ø.

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] ) 1. d ~ b ( d [b] ) 2. 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. Copyright © Peter Cappello 6

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

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) Copyright © Peter Cappello 7

Exercise 20 • Let P be the set of people who visited web page

Exercise 20 • Let P be the set of people who visited web page W. • Let R be a relation on P: x. Ry 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. Copyright © Peter Cappello 8

Exercise 20 continued • That is, x. Ry means s( x ) = s(

Exercise 20 continued • That is, x. Ry means s( x ) = s( y ). • x x. Rx: R is reflexive. Since x s( x ) = s( x ). • x y ( x. Ry y. Rx ): R is symmetric. Since s( x ) = s( y ) s (y ) = s( x ). • x y z ( ( x. Ry y. Rz ) x. Rz ): R is transitive. Since ( s( x ) = s( y ) = s( z ) ) s( x ) = s( z ). • Therefore, R is an equivalence relation. Copyright © Peter Cappello 9

Exercise 30 What are the equivalence classes of the bit strings for the equivalence

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 x. Ry such that x agrees with y on the left 3 bits (e. g. , 10111 ~ 101000). a) 010 b) 1011 c) 11111 d) 0101 Copyright © Peter Cappello 10

Exercise 30 Answer • 010 (answer: all strings that begin with 010) • 1011

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) • 0101 (answer: all strings that begin with 010) Copyright © Peter Cappello 2011 11

Exercise 40 a) What is the equivalence class of (1, 2) with respect to

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. Copyright © Peter Cappello 12

Exercise 40 a) Answer ( a, b ) ~ ( c, d ) ad

Exercise 40 a) Answer ( 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 }. Copyright © Peter Cappello 13

Exercise 40 continued b) Interpret the equivalence classes of the equivalence relation R in

Exercise 40 continued b) Interpret the equivalence classes of the equivalence relation R in Exercise 16. Copyright © Peter Cappello 14

Exercise 40 continued b) Interpret the equivalence classes of the equivalence relation R in

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. ) Copyright © Peter Cappello 15

Exercise 50 • A partition P’ is a refinement of partition P when x

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. Copyright © Peter Cappello 16

Exercise 50 continued It suffices to note that: Every county is contained within its

Exercise 50 continued It suffices to note that: Every county is contained within its state: No county spans 2 states. Copyright © Peter Cappello 17

Exercise 62 Determine the number of equivalent relations on a set with 4 elements

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? Copyright © Peter Cappello 18

End 8. 5 Copyright © Peter Cappello 19

End 8. 5 Copyright © Peter Cappello 19

10 Suppose A & R is an equivalence relation on A. Show f X

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 • – X = { [a] | [a] is an equivalence class of R } – a f (a ) = [a]. Then, a b a ~ b f( a ) = [a] = [b] = f( b ). Copyright © Peter Cappello 20