Understanding the Transitive, Reflexive, and Symmetric Properties

Date: 06/30/2008 at 21:20:22
From: Heather
Subject: Equivalence Relations

I am trying to help a person studying to be an elementary school 
teacher.  I am helping her with a math class, however I have gotten 
stuck on the transitive, reflexive and symmetrical properties.  For 
example: We need to classify each of the following math relationships 
on the specific sets as reflexive, symmetrical, or transitive. 
Determine whether each is an equivalence.

a) Is not equal to (counting numbers)
b) is less than (counting numbers)
c) has the same area as (triangles)

I understand what each property means, and I think I understand it 
when there is an actual set of numbers (such as {(1,1), (2, 3) etc}. 
I have run into some questions that seem more vague to me and am 
having a hard time applying the properties.  I guess I'm most 
confused because I'm not sure I understand what they mean with the 
given set. 

Going with letter a: When I see this, I think that for not equal to it
would not be reflexive since it says not equal to, it might be
symmetrical because if a is not equal to counting numbers then 
counting numbers are not equal to a, and it would not be transitive
because to be transitive a is equivalent to b and b is equivalent to c
so a is equivalent to c and I don't see how I could get 3 sets of
numbers from this problem.

Date: 06/30/2008 at 23:14:18
From: Doctor Peterson
Subject: Re: Equivalence Relations

Hi, Heather.

It looks like you are being confused by the fact that the properties
are often stated too briefly, not making clear what the x, y, and z
(or whatever letters are used) refer to.  Here's a statement of the
definitions:

  Relation R on a set S is reflexive if for any x in S, x R x.

  Relation R on a set S is symmetric if for any x and y in S
  such that x R y, it is also true that y R x.

  Relation R on a set S is transitive if for any x, y, and z in S
  such that x R y and y R z, it is also true that x R z.

Note that my x, y, and z are elements of the set S on which the
relation is defined.  They are not entire sets, or anything like that.
Also, what is said must be true for ANY x, y, and z.

(By the way, what I'm writing as x R y should be read not as "x is
equivalent to y" but as "x is related to y by R"; we don't know yet
that R is an equivalence relation, only that it is a relation.  You 
may be taught a different notation; let me know and I can rewrite 
this so it looks more familiar.)

Let's look at example (a):

  Is not equal to (counting numbers)

I'm assuming that "counting numbers" means the natural numbers 1, 2,
3, ... , though that doesn't make much difference here.

First, is this relation reflexive over the counting numbers?  By the
definition, we are asking whether, for ANY counting number x, it is
true that

  x is not equal to x

Clearly, this is NEVER true, so the relation is not reflexive.  Some
relations fail to be reflexive only because SOME numbers are not
"related" to themselves, though others are.

Is it symmetric?  Here we are asking whether, for ANY TWO counting
numbers x and y for which

  x is not equal to y

it is ALWAYS true that

  y is not equal to x

Well, we know very well that those two statements mean the same thing;
the fact that, say, 3 is not equal to 5 means the same as that 5 is
not equal to 3.  So it IS symmetric.

How about the transitive property?  That says that for ANY THREE
counting numbers x, y, and z for which

  x is not equal to y, and y is not equal to z

then it MUST be true that

  x is not equal to z

For example, it is true that

  3 is not equal to 5, and 5 is not equal to 7

and it is also true that

  3 is not equal to 7

So that example fits; and many others will too.  But is this ALWAYS
true?  Well, nothing in the definition says that x, y, and z all have
to be different elements; how about this case?

  3 is not equal to 5, and 5 is not equal to 3

Is it true that 3 is not equal to 3?  NO!

It can be hard sometimes to find a "counterexample" like this; you
have to think carefully about what it might take to make the statement
false.  In this case, I asked myself for an example of x and z that
are NOT "not equal", so the conclusion of the definition would be
false, and then chose a y that would make the condition true.

So we've found that this relation is symmetric, but not reflexive or
transitive.  It is not an equivalence relation.  That shouldn't be too
surprising; we wouldn't expect "not equals" to behave the same as
"equals" (which is what "equivalence relation" means); it's more
opposite to "equals".  What's interesting is to see in what ways it IS
the same, and in what ways it is not.

If you have any further questions, feel free to write back.


- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 07/03/2008 at 18:03:55
From: Heather
Subject: Thank you (Equivalence Relations)

Thank you so much!  I wish the book had put it that way, the other
problems made so much more sense after your explanation.  Thanks again!