1. ## relations help (2)

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

2. 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.

3. 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?

4. Originally Posted by jvignacio
Is that like saying the next full integer down?
Exactly.

5. 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?

6. 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.)

7. 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?

8. 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?

9. 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..

10. 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.

11. 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...

12. 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.)

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

14. Originally Posted by jvignacio
intervals $\langle n,n+1)$ , $n \in \mathbb{N}$ ?
Exactly.

15. Originally Posted by kompik
Exactly.
mate thanks alot for the help. Really appreciate your time.

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