Download Section 3.3 Equivalence Relation

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Section 3.3
Equivalence Relations
Section 3.3 Equivalence Relations
Relations
Purpose of Section To introduce the concept of an equivalence relation and
show how it subdivides or partitions a set into distinct categories.
Introduction
Classifying objects and placing similar objects into groups provides a way
to organize information and focus attention on the similarities of like objects
and not on the dissimilarities of dislike objects. Mathematicians have been
classifying objects for millennia. Lines in the plane can be subdivided into
groups of parallel lines. Lines in a given group many not be the same line, but
they are considered equivalent insofar as our classification is concerned.
Triangles can also be classified as being congruent1 or noncongruent. In
modular arithmetic we say two integers are equivalent if they have the same
remainder when divided by a given number, say 7 for example. Here, the
numbers … -12, -5, 2, 9, 16, 23, 30, … would be considered equivalent,
whereas and 3 and 5 are not. This classification based on division by 7
subdivides the entire set of natural numbers into seven distinct equivalence
classes depending on whether a number’s remainder is 0,1,2,3,4,5, or 6 when
divided by 7. So what are the properties a relation must have in order that it
subdivide set into distinct categories? Keep reading.
Equivalence Relation
The reader is already familiar with one equivalence relation, the relation
of two things being “equal”. That’s right " = " . However, since ambiguity
restricts us from using the equal sign to denote the equivalence relation, we
have adopted the nearby symbol2 " ≡ " .
So what properties does an
equivalence relation possess?
Definition A relation " ≡ " on a set A is an equivalence relation if for all x, y, z
in A , the following RST properties hold:
Reflexive:
x≡x
Symmetric: if x ≡ y, then y ≡ x
Transitive: if x ≡ y and y ≡ z , then x ≡ z .
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2
Recall that two triangles are congruent their corresponding sides and angles have the same measurements.
Another common notation for the equivalence relation is " ∼ " .
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Equivalence Relations
Note: Remember, a relation on a set A is a subset of all ordered pairs of A, so
keep in mind “ ≡ ” is a set. For example, x ≡ y means ( x, y ) ∈≡ . However, the
set notation looks awkward so we prefer to simply write x ≡ y .
Example 1 Equivalence Relations
The following relations are equivalence relations on the given sets.
▪
▪
▪
▪
x≡
x≡
x≡
x≡
y means equality (“=”) between numbers or sets.
y means “ x is congruent to y ” where x, y are triangles.
y means x ⇔ y for logical sentences x, y .
y means “x has the same birthday as y” where x and y
are people.
▪
x ≡ y means “ x differs from y by an integer multiple of 5”) on
the set of integers.
Example 2 Non Equivalence Relations
The following relations are not equivalence relations on the given sets since
at least one of the three RST conditions fails.
▪ x ≡ y means “x is in love with y” on the set of all people. (Not likely
symmetric; at least for two persons, one loves the other but not
Vice-versa.)
▪ x ≡ y means " x ≤ y " on the real numbers. (Not symmetric; 2 ≤ 3 does
not imply 3 ≤ 2 .)
▪ x ≡ y means “ x and y have a common factor greater than 1” on the
set of integers. (Not transitive; 2 and 6 have a common factor, 6 and 3
have a common factor, but 2 and 3 do not.)
▪ x ≡ y means “x ⊆ y” on a family of sets. (Not symmetric;
A ⊆ B ⇒ B ⊆ A .)
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Equivalence Relations
Equivalence Broadens the Concept of Equality
Partitioning of Sets into Equivalence Classes
The equivalence relation provides a way for mathematicians to subdivide
collections of objects into distinct subsets called partitions.
Definition: Let A be a set. A partition of A is a (finite or infinite) collection
{ A1 , A2 ,...} of nonempty subsets of A such that
▪ the union of all the Ai is A
▪ the sets in the collection are pairwise disjoint; that is Ai ∩ A j = ∅ for every
distinct pair Ai and A j .
Example 3 The following are partitions of a set.
▪ The even and odd integers are a partition the integers into two distinct
sets.
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Section 3.3
▪
Equivalence Relations
A collection of people partitioned into groups according to the first letter
in their surname.
▪ A set of students in a classroom partitioned into males and females. (If
there are no members of one sex, then the partition is the set itself.)
The following theorem lies at the heart of the importance of the equivalence
relation.
Theorem 1 Equivalence Classes
Let A be any set and " ≡ " an equivalence relation on A. For every x ∈ A ,
define a set, denoted by [ x ] , by
[ x ] = { y ∈ A : y ≡ x}
This set is called the equivalence
equivalence class of x and consists of all elements of A
equivalent to x . Under these conditions each element of the set A belongs to
one and only one equivalence class, which means the equivalence relation
partitions the set A into distinct subsets.
Proof:
First observe every element x ∈ A belongs to at least one equivalence
class since x ≡ x , which means x ∈ [ x ] . Also note the union of the equivalence
classes is A since every element x ∈ A belongs to some equivalence class.
Now show the equivalence classes are disjoint. We do this by showing if
two equivalence classes intersect, then they are the same equivalence class.
Let s, t ∈ A and assume that the equivalence classes [s ] and [ t ] which they
define, intersect. That is, there exists y ∈ [s ] ∩ [ t ] (as drawn in the diagram).
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Section 3.3
We now show [s ] = [ t ] .
Equivalence Relations
We first prove [s ] ⊆ [ t ] by letting x ∈ [s ] and showing
that x ∈ [ t ] . Letting x ∈ [s ] and using the symmetric and transitive properties
of the equivalence relation, we have
i) x ≡ s
ii ) but y ∈ [ s ] and so y ≡ s, hence s ≡ y
iii ) since x ≡ s and s ≡ y we have x ≡ y .
iv) but y ∈ [t ] and so y ≡ t
v) since x ≡ y and y ≡ t we have x ≡ t
This shows x ∈ [ t ] and hence [s ] ⊆ [ t ] .
The proof [s ] ⊇ [ t ] is similar and so
[s ] = [ t ] .
Significance of Theorem 1 (Partitioning Sets)
An equivalence relation ≡ defined on a set A , partitions the set into
disjoint subsets, called equivalence classes.
classes The elements of each equivalent
class are equivalent to each other, but not to any element in a different
equivalence class. One can think of the equivalence relation as a kind of
generalized equality where objects in a given class are “equal.” The set of
equivalences classes of A partitioned by ≡ is called the quotient set of A
modulo ≡ , and denoted by A/ ≡ . See Figure 1.
Partitioning of A into Equivalence Classes
Figure 1
Example 4 (Modular Arithmetic)
Let A = be the integers. We define two numbers x, y as equivalent if
they have the same remainder when divided by a given number, say 5, and we
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Section 3.3
Equivalence Relations
write this as x ≡ y ( mod 5 ) . Show this relation is an equivalence relation and
find the equivalence classes of this relation.
Solution
▪ reflexive:
x ≡ x ( mod 5) since 5 divides x − x = 0 .
▪ symmetric: If x ≡ y ( mod 5 ) then 5 divides x − y . This means there
exists an integer k such that x − y = 5k . But this implies
y − x = −5k = 5 ( −k ) which means 5 divides y − x . Hence y ≡ x ( mod 5) .
Hence ≡ is a symmetric relation.
▪
transitive: If x ≡ y ( mod 5) and y ≡ z ( mod 5) , then 5 divides
x − y and 5 divides y − z . Hence, there exist integers k1 , k2 that
satisfy x − y = 5k1 and y − z = 5k2 . Adding these equations yields
( x − y ) + ( y − z ) = 5k1 + 5k2
or
x − z = 5 ( k1 + k2 ) = 5k3
which shows that 5 divides x − z or x ≡ z ( mod 5 ) . Hence ≡ is a
transitive relation.
The above equivalence relation divides the integers into the equivalence
classes, where each equivalence class consists of integers with remainders of
0,1,2,3,4 when divided by 5. We give the numbers 0,1,2,3,4 special status and
assign them as “representatives of each equivalence class3 listed below.
Table
Table 1 Equivalence Classes for ≡ mod ( 5 )
[ 0] = {5n : n ∈ } = { − 10, − 5, 0, 5, 10 }
[1] = {5n + 1: n ∈ } = { − 9, − 4, 1, 6, 11 }
[ 2] = {5n + 2 : n ∈ } = { − 8, − 3, 2, 7, 12 }
[3] = {5n + 3 : n ∈ } = { − 7, − 2, 3, 8, 13 }
[ 4] = {5n + 4 : n ∈ } = { − 6, − 1, 4, 9, 14 }
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In number theory these equivalence classes are also called residue classes.
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Section 3.3
Equivalence Relations
In this example, the quotient set / ≡ (set of equivalence classes) would be
/ ≡ = {[ 0] , [1] , [ 2] , [3] , [ 4]}
Note: Some people do not understand how the remainder of −3 / 5 can be 2.
Remainders are defined as non negative integers, so we write
−3 / 5 = ( −5 + 2 ) / 5 = −1 + 2 / 5 .
Modular Arithmetic:
Arithmetic: Modular arithmetic occurs in many areas of mathematics
as well as application in our daily lives. Instead of working with an infinite
number of integers, it is only necessary to work with a few. The smallest
number is mod ( 2 ) , which subdivides the integers into two types, even and odd
integers. Public key cryptography allows computers to do exact arithmetic
mod ( n ) , where n is the product of two prime numbers each 100 digits long.
Example 5 (Equivalence Classes in the Plane)
Plane)
The Cartesian product A = × defines points in the first quadrant of
the Cartesian plane with integer coordinates. Define a relationship between
( a, b ) and ( c, d ) as
( a , b ) ≡ ( c, d ) ⇔ b − a = d − c .
Given that this relation defines an equivalence relation (See Problem 7), find
its equivalence classes.
Solution
This relation classifies points
( x, y )
as equivalent if their y coordinate
minus their x coordinate are equal. This means two points (with positive
integer coefficients), are equivalent if they both lie on 45 degree lines
y = x + n , where n is an integer. Each value of n defines a different
equivalence class. Some typical equivalence classes are
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Section 3.3
Equivalence Relations
Table 3 Equivalence Classes
...
...
( 2, 0 )  = {(2, 0), (3,1), (4, 2),...}
(1, 0 )  = {(1, 0), (2,1), (3, 2),...}
( 0, 0 )  = {(0, 0), (1,1), (2, 2),...}
( 0,1)  = {(0,1), (1, 2), (2,3),...}
( 0, 2 )  = {(0, 2), (1,3), (2, 4),...}
( n = −2 )
( n = −1)
( n = 0)
( n = 1)
( n = 2)
These equivalence classes are illustrated in Figure 2 as points with integer
coordinates lying on 45 degree lines in the first quadrant.
Equivalence Classes as Grid Points on Lines y = x + n
Figure 2
Margin Note:
Note: “ a is equivalent to b ” means a is equal to b , not in every
respect, but as far as a particular property is concerned.
Graph of an Equivalence Relation
Relation
The graph of the relation
R = {(1,1) , ( 2, 2 ) , ( 3,3)( 4, 4 ) , (1,3) , ( 3,1) , ( 3, 4 ) , ( 4,3)}
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Section 3.3
Equivalence Relations
on the set A = {1, 2,3, 4} is shown in Figure 3. It is an easy matter to verify that
R is an equivalence relation. What are the equivalence classes of this
relation?
Solution
The relation R says
1 ≡ 1, 2 ≡ 2, 3 ≡ 3, 4 ≡ 4, 1 ≡ 3, 3 ≡ 1, 3 ≡ 4, 4 ≡ 3
Using the transitivity property, we can show 1 ≡ 3 ≡ 4 , and that the number 2 is
only equivalent to itself. Hence the set A has been subdivided into two
equivalence classes {1,3, 4} and {2} . Note that the two equivalence classes
are disjoint and their union is A.
Graph of an Equivalence Relation
Figure 3
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Equivalence Relations
Problems
1.
Let A be the set of students at a university and let x and y be students; i.e.
elements of A . Tell if the following relations on A are equivalent relations.
a)
b)
c)
d)
e)
x
x
x
x
x
is related to y
is related to y
is related to y
is related to y
is related to y
iff
iff
iff
iff
iff
x and y
x and y
x and y
x and y
x and y
have the same major.
have the GPA.
are from the same country.
have the same major.
2. (Equivalence Relations?) Tell if the following relations R are equivalence
relations on a set A . If the relation is an equivalence relation, find the
equivalence classes.
a) xRy iff y = x 2
( A = )
b) xRy iff x is a factor of y
( A = )
c) xRy iff x and y are multiples of 5
( A = )
3. (Modular Arithmetic) Given the set A = {1, 2,3, 4,5, 6, 7,8} and for x, y ∈ A we
have x ≡ y ( mod 4 ) , i.e. 4 divides x − y .
Prove that ≡ is an equivalence
relation and find the equivalence classes of the relation.
3. (True or False)
a) 8 ≡ 3 ( mod 5 )
b) 12 ≡ 1( mod 6 )
c) 21 ≡ 0 ( mod 3)
d) 10 ≡ 10 ( mod12 )
e) 99 ≡ 9 ( mod 9 )
4. (Defining New Equivalence Relations) Let " ∼ " and " ≈ " be two equivalence
relations on a set. Define two more relations by
a b ⇔ ( a ∼ b) ∨ ( a ≈ b)
a b ⇔ ( a ∼ b) ∧ ( a ≈ b)
Determine if " " and " " are equivalence relations.
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Section 3.3
Equivalence Relations
5. (An Old Favorite) The equals relation " = " is the most familiar equivalence
relation. What are the equivalence classes of the equals relation on the set
A = {1, 2,3, 4,5} ?
6.. (Equivalence Relation) The Cartesian product A = × defines the points
in the first quadrant with integer coordinates. Show the relationship between
( a, b ) and ( c, d ) defined by
( a , b ) ≡ ( c, d ) ⇔ b − a = d − c
is an equivalence relation.
7. (Graph of an Equivalence Relation) Figure 4 shows an equivalence relation
on the set A = {1, 2,3, 4} . What are the equivalence classes?
Graph of an Equivalence Relation
Figure 4
8. (Projective Plane)
Define A = 2 −
{( 0, 0 )}
the set of all points in the plane
minus the origin. Define a relation between two points
( x, y )
and
( x′, y′ )
by
saying they are related if and only if they lie on the same line passing through
the origin.
Show that this relation is an equivalence relation and draw a
graphical representation of the equivalence classes by picking a
representative from each class. The space of all equivalence classes under
this relation is called the projective plane.
plane
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Equivalence Relations
9. (Equivalence Classes in Logic) Define an equivalence relation on logical
sentences by saying two sentences are equivalent if they have the same truth
value. Find the equivalence classes in the following collection of sentences.
1+ 2 = 3
3<5
2|7
x 2 < 0 for some real number.
sin 2 x + cos 2 x = 1
Georg Cantor was born in 1845.
Leopold Kronecker was a big fan of Cantor.
Cantor's theorem guarantees larger and larger infinite sets.
10.
(Similar Matrices)
M such that MAM
−1
Two matrices A, B if there is an invertible matrix
=B.
a) Show that similarity of matrices is an equivalence relation.
b) Let
1 1
1 0
A=
, M =


 4 1
1 1 
find B . What common properties do A, B share?
11. (Equivalence Relation on ) Let f : → . Define a relation on by
a ≡ b ⇔ f ( a ) = f (b )
a) Show ≡ is an equivalence relation.
b) What are the equivalence classes for f ( x ) = x 2 ?
c) What are the equivalence classes for f ( x ) = sin x ?
12. (Arithmetic in Modular Arithmetic) Suppose
a ≡ c ( mod 5 )
b ≡ d ( mod 5 )
Show
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Section 3.3
Equivalence Relations
a) a + b ≡ c + d ( mod 5 )
b) a − b ≡ c − d ( mod 5)
c) ab ≡ cd ( mod 5)
13. (Equivalent Angles)
Angles) We say two angles x, y are related if and only if
xRy ⇔ sin 2 x + cos 2 y = 1
Show that R is an equivalence relation and identify the equivalence classes.
14.
(Moduli in Modular Arithmetic) Note that 2 ⋅ 2 = 0 ( mod 4 ) but neither factor
is zero. What is 2 ⋅ 4 modulo 8? Is there any product a ⋅ b with neither factor
0, but a ⋅ b ≡ 0 ( mod 3) ? Can you form a conjecture about the moduli related to
this phenomenon?