Final part of determining an equivalence relation

In summary, Homework Equations state that if a,b are elements of the real numbers and b,c are elements of the real numbers, then (a,c) is also an element of the real numbers.
  • #1
kathrynag
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0

Homework Statement


For a,b elements of the real numbers, define a~b if [tex]\left|a-b\right|[/tex][tex]\leq[/tex]
1. Determine if we have an equivalence relation.


Homework Equations





The Attempt at a Solution


I've already done the first two parts of determining. it's only the last part that I'm having trouble with, so i will omit the first two parts.
We bwant to determine if for all a,b,c elements of S, if (a,b) is an element of R and (b,c) is an element of R, then (a,c) is an element of R.
We know [tex]\left|a-b\right|[/tex][tex]\leq[/tex]1 and [tex]\left|b-c\right|[/tex][tex]\leq[/tex]1.
This means a-b[tex]\leq[/tex]1,b-a[tex]\leq[/tex]1, b-c[tex]\leq[/tex]1, and c-b[tex]\leq[/tex]1.
Furthermore, a[tex]\leq[/tex]1+b, b[tex]\leq[/tex]1+a, b[tex]\leq[/tex]1+c, c[tex]\leq[/tex]b+1.
I know from this, I need to be able to show a-c[tex]\leq[/tex]1 and c-a[tex]\leq[/tex]1 in order to have an equivalence relation.
That's what I don't see right off hand is how to get that.
 
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  • #2
How about a = 0, b = 1, c = 2?
 
  • #3
But that's not how to determine an equivalence relation. I can't just pick numbers because it has to work for all a, b and c.
 
  • #4
kathrynag said:
it has to work for all a, b and c.

Exactly.

You might want to carefully re-read the question :)
 
  • #5
Could I do something like this:
Ok if c[tex]\leq[/tex]1+b and a[tex]\leq[/tex]1+b.
We have c[tex]\leq[/tex]a or c-a[tex]\leq[/tex]0.
If a[tex]\leq[/tex]1+b and c[tex]\leq[/tex]1+b, we can say a[tex]\leq[/tex]c or a-c[tex]\leq[/tex]0.
Could I do something like that since that implies [tex]\left|a-c\right|[/tex][tex]\leq[/tex]0, but we wanted [tex]\left|a-c\right|[/tex][tex]\leq[/tex]1. Thus not an equivalence relation.
 
  • #6
Why do you want to do it the hard way?
If it is an equivalence relation, |a - b| <= 1 and |b - c| <= 1 should imply |a - c| <= 1 for all a, b and c.
So if you can find just one pair (a, b, c) for which it's not true, you are done, right?
 
  • #7
CompuChip said:
Why do you want to do it the hard way?
If it is an equivalence relation, |a - b| <= 1 and |b - c| <= 1 should imply |a - c| <= 1 for all a, b and c.
So if you can find just one pair (a, b, c) for which it's not true, you are done, right?

Well, I know my professor would not want to just see a pair since he is a very strict grader.
 
  • #8
hi kathrynag! :smile:

(have a ≤) :wink:
kathrynag said:
Well, I know my professor would not want to just see a pair since he is a very strict grader.

honestly, compuchip is right …

your professor asked you to determine whether it is an equivalence relation …

if it is, then of course you have to prove it strictly … as you say, you can't just pick numbers because it has to work for all a, b and c

but if it isn't, then you prove that simply by finding one a b and c for which it doesn't work :wink:
 
  • #9
Did you read this Wikipedia page I linked you to earlier (actually, the first paragraph of the introduction will do)?
 

Related to Final part of determining an equivalence relation

1. What is an equivalence relation?

An equivalence relation is a mathematical concept that defines a relationship between two elements in a set. It is a symmetric, reflexive, and transitive relation, meaning that it follows certain properties and rules.

2. What is the final part of determining an equivalence relation?

The final part of determining an equivalence relation is to check for transitivity. This means that for any three elements in the set, if the first element is related to the second and the second element is related to the third, then the first element must also be related to the third.

3. Why is transitivity important in determining an equivalence relation?

Transitivity is important because it ensures that the relation is consistent and follows a logical pattern. Without transitivity, the relation may not be considered an equivalence relation.

4. How do you prove transitivity in an equivalence relation?

To prove transitivity, you can use a direct proof by showing that for any three elements in the set, if the first two are related and the second two are related, then the first and third elements must also be related. You can also use a proof by contradiction or a proof by contrapositive.

5. Can an equivalence relation have more than one final part?

No, an equivalence relation can only have one final part, which is transitivity. This is because the other two properties of an equivalence relation, reflexivity and symmetry, can be easily checked and do not require a specific "final" step.

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