relations help (2)

**jvignacio** · April 18th 2010, 03:41 AM

hey guys, question:

Determine whether the following relation is a reflexive, symmetric or transitive. If any is an equivalence relation, describe its equivalence classes.

xRy iff x has the same integer part as y, on

<----- I need help on what this relation is asking? thanks

**kompik** · April 18th 2010, 03:57 AM

Originally Posted by jvignacio

hey guys, question:

Determine whether the following relation is a reflexive, symmetric or transitive. If any is an equivalence relation, describe its equivalence classes.

xRy iff x has the same integer part as y, on

<----- I need help on what this relation is asking? thanks

The integer part of x is the integer n such that $n\le x < n+1$ . (There is exactly one such integer for each real x.) It is usually denoted by [x].

For example:
[3.14]=3
[0.52]=0
[1]=1
[-1.26]=-2

So 2 and 2.5 are in relation, since [2]=2=[2.5], but 1.5 and 2.5 are not.

Hope this helps.

**jvignacio** · April 18th 2010, 04:04 AM

Originally Posted by kompik

the integer part of x is the integer n such that $n\le x < n+1$ . (there is exactly one such integer for each real x.) it is usually denoted by [x].

For example:
[3.14]=3
[0.52]=0
[1]=1
[-1.26]=-2

so 2 and 2.5 are in relation, since [2]=2=[2.5], but 1.5 and 2.5 are not.

Hope this helps.

Is that like saying the next full integer down?

**kompik** · April 18th 2010, 04:07 AM

Originally Posted by jvignacio

Is that like saying the next full integer down?

Exactly.

**jvignacio** · April 18th 2010, 04:12 AM

Originally Posted by kompik

Exactly.

So in this case,

- It IS reflexive since x has the same integer part as x.
- It IS symmetric since if x has the same integer part as y then y has the same integer part as x.
- It IS transitive since if x has the same integer part as y and if y has the same integer part as z then x has the same integer part as z.

correct?

**kompik** · April 18th 2010, 04:16 AM

Originally Posted by jvignacio

So in this case,

- It IS reflexive since x has the same integer part as x.
- It IS symmetric since if x has the same integer part as y then y has the same integer part as x.
- It IS transitive since if x has the same integer part as y and if y has the same integer part as z then x has the same integer part as z.

correct?

Yes, it is an equivalence relation.

Another way to see this is to notice that you're in fact given a decomposition of R and every decomposition gives you an equivalence relation. (But if you haven't heard much about the correspondence between equivalences and decompositions at your lessons, you should perhaps ignore this comment.)

**jvignacio** · April 18th 2010, 04:20 AM

Originally Posted by kompik

Yes, it is an equivalence relation.

Another way to see this is to notice that you're in fact given a decomposition of R and every decomposition gives you an equivalence relation. (But if you haven't heard much about the correspondence between equivalences and decompositions at your lessons, you should perhaps ignore this comment.)

Yeah ive never herd of that... Not yet anyway. What would the equivalence class be in this case now that its an equivalence relation? All numbers?

**kompik** · April 18th 2010, 04:23 AM

Originally Posted by jvignacio

Yeah ive never herd of that... Not yet anyway. What would the equivalence class be in this case now that its an equivalence relation? All numbers?

No.
Take, for instance, the equivalent class of 3.14. It contains all numbers such that x R 3.14.
This is equivalent to
[x]=[3.14]=3.
And what are the numbers such that [x]=3? Precisely the numbers from the interval $\langle 3,4)$ , right?
Can you find the remaining classes?

**jvignacio** · April 18th 2010, 04:48 AM

Originally Posted by kompik

No.
Take, for instance, the equivalent class of 3.14. It contains all numbers such that x R 3.14.
This is equivalent to
[x]=[3.14]=3.
And what are the numbers such that [x]=3? Precisely the numbers from the interval $\lange 3,4)$ , right?
Can you find the remaining classes?

Sorry whats in the interval for [x]=3? theres a latex error..

**kompik** · April 18th 2010, 04:49 AM

Originally Posted by jvignacio

Sorry whats in the interval for [x]=3? theres a latex error..

Sorry, did not notice that. I've edited my post.

**jvignacio** · April 18th 2010, 04:56 AM

Originally Posted by kompik

No.
Take, for instance, the equivalent class of 3.14. It contains all numbers such that x R 3.14.
This is equivalent to
[x]=[3.14]=3.
And what are the numbers such that [x]=3? Precisely the numbers from the interval $\langle 3,4)$ , right?
Can you find the remaining classes?

Ok I understand the equivalent class of 3.14 and the numbers such that [x]=3 are all numbers between 3 and 4 but Which remaining classes are you referring too? Since my question is a x and y question, how can I write this in a general form. If you get what I mean...

**kompik** · April 18th 2010, 05:00 AM

Originally Posted by jvignacio

Ok I understand the equivalent class of 3.14 and the numbers such that [x]=3 are all numbers between 3 and 4 but Which remaining classes are you referring too? Since my question is a x and y question, how can I write this in a general form. If you get what I mean...

If I were your teacher, I would expect answer to be something like:
The equivalent classes are intervals .... (fill in the dots) for $n\in\mathbb{N}$ .
(I guess the example with 3.14 might help you to see what to fill in.)

**jvignacio** · April 18th 2010, 05:05 AM

Originally Posted by kompik

If I were your teacher, I would expect answer to be something like:
The equivalent classes are intervals .... (fill in the dots) for $n\in\mathbb{N}$ .
(I guess the example with 3.14 might help you to see what to fill in.)

intervals $\langle n,n+1)$ , $n \in \mathbb{N}$ ?

**kompik** · April 18th 2010, 05:06 AM

Originally Posted by jvignacio

intervals $\langle n,n+1)$ , $n \in \mathbb{N}$ ?

Exactly.

**jvignacio** · April 18th 2010, 05:14 AM

Originally Posted by kompik

Exactly.

mate thanks alot for the help. Really appreciate your time.

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