very confused since I missed class...relations

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!

Equivalence Class: Notation

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 ... Quote:

Originally Posted by

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

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

... 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 $\displaystyle /$ is being used in two different ways when a relation $\displaystyle R$ is defined on a set $\displaystyle S$:

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

Perhaps Plato would like to comment on this?

Grandad