Thomas Meade
September 18, 2008
MAE301
Class Notes: 9/16/08
Homework review in response to question 1(e):
The question stated: Give an example of a function
that is surjective but not injective.
Through class discussion the following function was suggested as a possible solution to this exercise:
where if
-4
-3
-2
-1
0
1
2
3
4
In order to prove this function fulfills the examples
criteria of being surjective but not injective there
are two checks that must be performed:
1. Proof that
is not injective:
-2
-2
-1
-1
0
1
1
2
2
Because two different values of yield the
same value, the function is NOT injective.
2. Proof that
is surjective:
Choose a value y codomain
let
then
where
=y
Hence:
-----------------------------------------Moving on---------------------------------------------Equivalence Relations
In order to talk about equivalence relations we must first define what is meant by the term “relation”. A
relation on a set
is determined by a property which may or may not be satisfied by each ordered pair of
elements on the set.
-
If an ordered pair of elements ( ) satisfies the property then we say that the elements are
related and write
to indicate this.
-
To indicate that the ordered pair ( , ) do not satisfy the property we write
the elements are not related.
, meaning
As defined by Keith, an equivalence relation, on a nonempty set , is a relation that satisfies the
reflexive, symmetric and transitive properties
.
Reflexivity: for all
Symmetry: if
such that
Transitivity: if
such that
When discussing equivalence relations there are 2 key items that must be defined; The first item being the
set, and the second being the relation.
1. The set defines the type of information that is going to be worked with. (I.E.
fractions, etc…)
2. The relation is the property that we are testing for. (
amount, etc…)
difference of a predetermined
An equivalence relation will always partition the set into disjoint subsets, known as equivalence classes.
A partition of the set
is a set of subsets such that:
-
The subsets are non-empty
-
The subsets are disjoint
-
The subsets cover
Definition: Suppose that is an equivalence relation on a nonempty set . For each
we
define the equivalence class of to be the elements of which are equivalent to . We denote
this class by . Thus,
(make note that each equivalence class is a subset of )
Example 1: Parity Equivalence Relation
In
we say
if
and
are of the same parity. Parity is defined as “the property of an
integer with respect to being odd or even”. In order to determine if there is an equivalence relation based
on the parity of integers reflexivity, symmetry and transitivity must be checked. Another good thing to
take note of is the set being worked with, the integers in this case, and also the relation, which is parity.
-
Reflexivity:
always has the same parity as itself therefore:
-
Symmetry: if
-
Transitivity: if
have the same parity and
have the same parity, written as
have the same parity then:
have the same parity, then
With these three checks complete we conclude that there is an equivalence relation based on
parity, and the equivalence classes are: even and odd, or ( , ).
Example 2: “Days of the Week” Equivalence Relation
In the set of “All Days of the Week”, (any Mon, any Tues, any Wed, any Thurs, any Fri, any Sat, any
Sun), given 2 day, say
and
, we say
if the distance from
to
is a multiple of 7 days.
Question: determine if there exists an equivalence relation for the days of the week, based on the
information provided.
SET: the set of all days, RELATION: The distance between days is a multiple of 7
Reflectivity: since the distance from
say
Symmetry: if
distance from
to
to
is 0, which is a multiple of 7 because 7*0=0, we
then the distance from
would be -7k, so
to
is 7k, for some integer k. similarly the
and therefore
Transitivity: if
and
then the distance from
to
is 7
is 7 , then the distance from
to
is 7( + ). Therefore.
, and the distance from
In this case, the equivalence classes are the days of the week, as defined above.
Answer: Based on the set and relation provided by the question, there DOES exist an
equivalence relation for the days. The equivalence classes are (Mon, Tues, Wed, Thurs, Fri, Sat,
Sun).
Example 3: Divisor of 2 Equivalence Relation
In the set of ,
are said to be equivalent if:
[meaning 2 divides the difference between
]
Question: Does there exist an equivalence relation based on this information?
SET: Integers, RELATION: 2 /
Reflectivity:
,
Symmetry:
suppose
That means: 2 / (
So:
And:
Therefore: 2 /
Hence:
and 2 divides 0 therefore,
.
, for some
Transitivity:
suppose
Then: 2 / (
and 2 / (
So: (
= 2 and (
and
=2
, where
,
Then:
And:
Therefore: 2 /
Hence:
Answer: Yes, there does exist an equivalence relation based on a divisor of 2 for the difference
between 2 integers.
is an example of an equivalence, the question “what types of
After determining that 2 /
numbers would be in the set that satisfy this equivalence relation?” was asked by our professor.
-
In response one student replied, any pair of numbers whose difference is a multiple of
2 would be in the set.
-
As a follow on, a second student added that all even numbers are in one equivalence
class together. As the proof they stated:
Let
and
Then:
Therefore: 2 /
So:
and
-
be used to denote 2 even integers, where
=
, which is even.
,
are in the same equivalence class.
.
It was further stated by a third student that all odd numbers are a multiple of 2 away
from each other, hence it would be applicable to them as well.
Let
+1 and
+1 be used to denote 2 odd integers, where
Then:
=
Therefore: 2 /
,
So:
and
are in the same equivalence class.
Example 4: Using Modular Arithmetic
Question: In
it is said that
if 5 /
. Prove this is an equivalence relation. (as a hint
from the professor, it was recommended that we perform the same steps we used in the previous
exercise but instead of factoring out a 2, you factor out a 5).
-
While discussing this question amongst the class, one student raised the question as to
whether or not a modular function could be used.
Definition: Two integers a and b are said to be congruent modulo m when
is divisible by , i.e. there exists an integer such that
. We write
and in the case of our example we
would say
-
By using this approach to solve the equivalence relation, our equivalence classes
would be all the possible values of
. While using
those possible classes are
0,1,2,3,4 and we would denote those classes in the form of , or [0], or even
(which is mod 5 specific)
-
5ℤ
is used to denote the set of all of the equivalence classes.
The use of modular arithmetic is first introduced to students in the form of multiplication and
addition tables, similar to the one below:
+
0
1
2
3
4
0
0
1
2
3
4
1
1
2
3
4
0
2
2
3
4
0
1
3
3
4
0
1
2
4
4
0
1
2
3
This table is an example of a table that a student might see if learning about integers mod 5. The
highlighted 0 above is going to be used as an example of modular arithmetic. If you add the 1 (to
the far left) and the 4 (2 rows above), the sum is 5. However,
-
.
Claiming the set of classes has algorithmic structure from the chart, we can now
define addition by adding 2 sets together. To prove this we will attempt to add 2
numbers first and then find it’s congruence mod 5, and then try to add the classes the
2 numbers belong to, to verify our answer. For the addition of classes to be possible
we need to obtain the same answer for an addition problem regardless of which
representatives we choose.
Example 1 (using mod 5)
, we chose 9 and 14 because they are both
equivalent to
Now checking addition of classes we see that
, which is the same answer that we got in the
problem above. In this example addition is the same no matter how
it is represented.
This leads us to the definition
second sum)
=
(first ignoring the bar on the
Referring back to the question asked by Example 4, we want to show that
If:
and
Then:
=
That:
=
Proof: Take (
)–(
)
Which is equal to: (
)–(
By the initial assumption:
Then: 5 / (
) and 5 / (
So: 5 / [(
)+ (
)]
Rewritten as: 5 / [(
)-(
So: (
)
Therefore:
)
and
)
)]
While establishing equivalence classes for any mod function, the classes
will be all the possible values of from the equation
. Every
possible integer will be present in one of these classes. Also, you will be
able to form an equation for each class to determine what possible values
will fall in that class. (for the mod 5 function refer to the table below)
Class
Values
Representative
Equation (
(for the mod 5 function)
0, 5, 10, 15, …
1, 6, 11, 16, …
2, 7, 12, 17, …
3, 8, 13, 18, …
4, 9, 14, 19, …
(Note: if you use this same proof and replace the 5 with a variable, say m, it turns into the proof for all equivalent classes )
As a quick review, equivalence relations are composed of 3 parts:
1. They must have a defined set.
2. They must have a defined relation.
3. They must have equivalence classes.
Example 2 is a good example of these 3 necessities of an equivalence relation:
Set
Relation
Days
if the differ by 7k
days
Classes
(Mon, Tues, Wed,
Thurs, Fri, Sat, Sun)
Some other examples of equivalence relations found in middle school and high school
mathematics classes are:
n / b-d
Integers
Fractions:
(
if
)
(
)
such that ,
Triangles are another example of a group that has an equivalence relation. That relation
being congruency:
Check: if
Reflexivity:
Symmetry:
Transitivity: if
then
then
and
, then
Notice that in the cases of Days of the week there are infinitely many days but finitely many
equivalence classes. However for the integers, fractions and triangles:
-
For the set of integers, there are infinitely many integers but there are finitely many
equivalence classes, each containing infinitely many elements.
-
A fraction is an equivalence class with respect to the relation. Hence
stand for the pair
doesn’t
, it stands for the infinite set of all pairs equivalent to
.
Thus, there exist infinitely many equivalence classes, each containing infinitely many
elements per class.
-
Triangles are an example of an equivalence relation that consists of infinitely many
equivalence classes, because there are infinitely many possible triangles, and each
equivalence class contains infinitely many elements.
Extra Example: Parallel Lines
Two lines,
and
, are said to be parallel if they are not transverse and is denoted as
. By
definition, any line is parallel to itself. Prove that parallel lines form an equivalence relation:
Check: if 2 lines
then
Reflexivity:
therefore
Symmetry:
then
, therefore
Transitivity: if
and
, then
, therefore
Answer: Based on the reflexive, symmetric, and transitive properties, parallel lines do form an
equivalence relation, consisting of an infinite number of equivalence classes since there are an
infinite number of slopes of lines.
References
Parity. (2008). In Merriam-Webster Online Dictionary. Retrieved September 17, 2008,
from http://www.merriam-webster.com/dictionary/parity.
Eccles, Peter J. (2007). An Introduction to Mathematical Reasoning: numbers, sets and functions (9th printing).
New York: Cambridge University Press.
Insall, Matt and Weisstein, Eric W. "Modular Arithmetic." From MathWorld--A Wolfram Web Resource. Retrieved
September 17, 2008, http://mathworld.wolfram.com/ModularArithmetic.html.
Stony Brook Mathematics Department. (2006). From MAT 200 Course Notes on Geometry. Retrieved
September 17, 2008 from http://www.math.sunysb.edu/~sbsimon/200F08/Geometry.pdf
Coutinho,S., C. (1999). Mathematics of Ciphers: Number Theory and RSA Cryptography. Retrieved October 13,
2008, from http://books.google.com/books?id=3RfJN-2kPgwC&printsec=frontcover&dq=The+
Mathematics+of+Ciphers&ei=Uj73SP3cHpCoMsKTocMI#PPP1,M1