very confused since I missed class...relations

**zhupolongjoe** · March 20th 2009, 12:47 PM

Describe the partition for each of the following equivalence relations:

(c) for x, y belong to R, xRy iff sinx=siny
(d) For (x,y) and (u,v) belong to RxR (x,y)S(u,v) iff xy=uv=0 or xyuv>0
(e) For x,y belong to R, xTy iff [x]=[y] where [x] is defined to be the greatest integer in x.

Thanks for help on any of them!

**Grandad** · March 20th 2009, 01:43 PM

Hello zhupolongjoe

Originally Posted by zhupolongjoe

Describe the partition for each of the following equivalence relations:

(c) for x, y belong to R, xRy iff sinx=siny
(d) For (x,y) and (u,v) belong to RxR (x,y)S(u,v) iff xy=uv=0 or xyuv>0
(e) For x,y belong to R, xTy iff [x]=[y] where [x] is defined to be the greatest integer in x.

Thanks for help on any of them!

(c) Any given equivalence class will contain numbers whose sines are equal. For example, $\sin 0 = \sin \pm\pi =\sin \pm2\pi = \dots$ . So $0, \pm\pi, \pm2\pi, \dots$ will all be in the same equivalence class.

In general, if $\sin x = \sin y$ , then $x = n\pi +(-1)^ny$ . So all the numbers in a given equivalence class will be related in this way.

(d) Think about the rules you are given that make $(x,y)$ and $(u,v)$ related:

Either $xy = uv = 0$ . In other words, at least one of $x$ or $y$ and at least one of $u$ and $v$ must be zero.
Or $xyuv > 0$ . In other words, the product of all four numbers $x, y, u$ and $v$ is positive. So what does this tell you about the individual signs of $x, y, u$ and $v$ ?

Then use the fact that whenever two ordered pairs $(x,y)$ and $(u, v)$ are related by one or other of these rules, then they will be in the same equivalence class.

(e) Think of some numbers $x$ , $y$ for which $\lfloor x\rfloor =$ $\lfloor y\rfloor$ . E.g. $\lfloor 5\rfloor = \lfloor 5.1\rfloor = \lfloor 5.99\rfloor = \lfloor 5.99999\rfloor$ . So all these numbers will be in the same equivalence class.

Grandad

**zhupolongjoe** · March 20th 2009, 02:50 PM

You've been extremely helpful to me today. I really appreciate it. Thank you!

**Plato** · March 20th 2009, 03:47 PM

This post is simply to expand on the previous post.
Here are the representations of equivalence classes.

For part e: $x/R = \left[ x \right] = \left[ {\left\lfloor x \right\rfloor ,\left\lfloor x \right\rfloor + 1} \right)$ .

For part c: define a mapping $f:\mathbb{R} \mapsto \left[ {0,2\pi } \right]\text{ as }f(x) = \left[ {x - \left\lfloor {\frac{x}<br /> {{2\pi }}} \right\rfloor \left( {2\pi } \right)} \right]$ .
Now we can define the classes.
$x/R = \left[ x \right] = \left[ {f(x)} \right]$ .

**Grandad** · March 20th 2009, 11:06 PM

Hello everyone -

With reference to my previous criticism of the use of the 'quotient set' / notation (http://www.mathhelpforum.com/math-he...tml#post285435), I see that Plato is using it here ...

Originally Posted by Plato

...For part e: $x/R = \left[ x \right] = \left[ {\left\lfloor x \right\rfloor ,\left\lfloor x \right\rfloor + 1} \right)$ .

... $x/R = \left[ x \right] = \left[ {f(x)} \right]$ .

... to represent an equivalence class. This is a use of the notation with which I am not familiar. However, if it is accepted these days, then I withdraw my previous criticism.

So you need to understand that $/$ is being used in two different ways when a relation $R$ is defined on a set $S$ :

$S/R$ represents the quotient set, or the set of all the equivalence classes.
If $x \in S, x/R$ represents the single equivalence class that contains the element $x$ . Other notations that are (I think!) in more common use to represent this class are $[x]$ and $[x]_R$ .

Perhaps Plato would like to comment on this?

Grandad

**Plato** · March 21st 2009, 05:13 AM

Originally Posted by Grandad

Perhaps Plato would like to comment on this?

I just assumed that the two notations are in common use.
Equivalence relation - Wikipedia, the free encyclopedia

P.S. Both notations are found in the 1969 ed. of NAIVE SET THEORY by Paul Halmos.

**Grandad** · March 21st 2009, 06:35 AM

Originally Posted by Plato

I just assumed that the two notations are in common use.
Equivalence relation - Wikipedia, the free encyclopedia

P.S. Both notations are found in the 1969 ed. of NAIVE SET THEORY by Paul Halmos.

I have scoured many internet articles (including the Wikipedia one you quote) that mention the '/' notation and, although I have found many references to 'set / relation' to represent the quotient set, I have yet to find one that uses 'element / relation' to represent the equivalence class. Not that that proves it's not used in this way, of course.

But personally, I would steer clear of it, and stick to the widely accepted $[x]$ or $[x]_R$ to represent the equivalence class. In my experience, students often find relations confusing enough without using / to mean two different things!

Grandad

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