1. ## another equivalence relation..

Define the equivalence relation S by S = {(x,y)ͼAxA | 5 divides (x^2-y^2)}.
Determine the partition of A that is induced by S.

I am not exactly sure how to do this one but from what I gather from my textbook it seems to sorta start like this..

Given
$
S \cdot S = \{(x,y) \in AxA | 5 divides (x^2-y^2)\}
$

$
=\{(x,y) \in A^2 | (x^2-y^2) = 5k, \forall k \in Z\}
$

$
=\{(x,y) \in A^2 | (x^2 = 5k+y^2, \forall k \in Z\}
$

$
=\{(x,y) \in A^2 | y^2 = x^2-5k, \forall k \in Z\}
$

Am I doing this correctly? On the right track?

2. i got the final answers of..

X = {xͼA | All real numbers }
Y = {yͼA | -sqrt(5k) >= y sqrt(5k), For some integer of k}

dunno if that is correct..

3. ## Equivalence Classes

Hello Kitizhi
Originally Posted by Kitizhi
Define the equivalence relation S by S = {(x,y)ͼAxA | 5 divides (x^2-y^2)}.
Determine the partition of A that is induced by S.

I am not exactly sure how to do this one but from what I gather from my textbook it seems to sorta start like this..

Given
$
S \cdot S = \{(x,y) \in AxA | 5 divides (x^2-y^2)\}
$

$
=\{(x,y) \in A^2 | (x^2-y^2) = 5k, \forall k \in Z\}
$

$
=\{(x,y) \in A^2 | (x^2 = 5k+y^2, \forall k \in Z\}
$

$
=\{(x,y) \in A^2 | y^2 = x^2-5k, \forall k \in Z\}
$

Am I doing this correctly? On the right track?
Are you familiar with the modulo notation, to show the remainder when one integer is divided by another? If $x^2-y^2$ is a multiple of $5$, then $x^2-y^2 \equiv 0 \mod 5$

Now look at the various possible values of $x^2$, for $x \equiv 0, 1,2 , 3, 4 \mod 5$:

$x \equiv 0 \Rightarrow x^2 \equiv 0$

$x \equiv 1 \Rightarrow x^2 \equiv 1$

$x \equiv 2 \Rightarrow x^2 \equiv 4$

$x \equiv 3 \Rightarrow x^2 \equiv 4$

$x \equiv 4 \Rightarrow x^2 \equiv 1$

So $(x \equiv 0) \land (x^2-y^2 \equiv 0) \Rightarrow y \equiv 0 \mod 5$

And $\Big((x \equiv 1) \lor (x \equiv 4)\Big) \land (x^2-y^2 \equiv 0) \Rightarrow (y \equiv 1) \lor (y \equiv 4) \mod 5$

Do you understand the notation I've used here? Can you complete it, starting with $\Big((x \equiv 2) \lor (x \equiv 3)\Big) \land (x^2-y^2 \equiv 0) \Rightarrow ...$ and then using this to describe the equivalence classes?