# Thread: equivalence relation on a partition..help.

1. ## equivalence relation on a partition..help.

Lets A = {1,2,3,4,5,6,7,8,9,10} and let P be the partition of A given by
P = {{1,2},{3,4,5},{6,7},{8,10},{9}}

Give the equivalence relation R on A that is induced by P.

I am not sure if I am doing this write. Can someone tell me if this is correct or let me know if I am missing anything? Thanks.

R is reflexive iff For all x ͼ A, xRx.

P={(1,1),(2,2),(3,3),(4,4),(5,5),(6,6),(7,7),(8,8) ,(9,9),(10,10)}

R is symmetric iff For all x,y ͼ A, if xRy then yRx.

P={(1,2),(2,1),(3,4),(4,3),(4,5),(5,4),(5,3),(3,5) ,(6,7),(7,6),(8,10),(10,8)}

R is transitive iff For x,y,z ͼ A, if xRy and yRz then xRz.

(3,4)ͼA, (4,5)ͼA and so (3,5)ͼA.
(5,4)ͼA, (4,3)ͼA and so (5,3)ͼA.

I think this is all of the possible transitive possibilities but I also dunno if I have to list them all, I would assume so.

2. Here is the way it is done.
Suppose that $\displaystyle \mathbb{P} = \{A,B,C\}$ is a partition of the set $\displaystyle \mathcal{X}$ then the equivalence relation on $\displaystyle \mathcal{X}$ determined by $\displaystyle \mathbb{P}$ is $\displaystyle \left( {A \times A} \right) \cup \left( {B \times B} \right) \cup \left( {C \times C} \right)$.

Does that help?

3. not really...

I am not 100% what I am suppose to find exactly. It says to give the equivalence relation, so is it asking for the reflexive,symmetric, and transitive relation of this partition? Cause that is how I seemed to have answered it..

4. Originally Posted by Kitizhi
It says to give the equivalence relation, so is it asking for the reflexive,symmetric, and transitive relation of this partition? Cause that is how I seemed to have answered it..
Any relation is a set of ordered pairs.
Form each of these cross products.
$\displaystyle \begin{gathered} \left\{ {1,2} \right\} \times \left\{ {1,2} \right\} \hfill \\ \left\{ {3,4,5} \right\} \times \left\{ {3,4,5} \right\}\;\& \;\left\{ {6,7} \right\} \times \{ 6,7 \hfill \\ \left\{ {8,10} \right\} \times \left\{ {8,10} \right\}\;\& \;\left\{ 9 \right\} \times \left\{ 9 \right\} \hfill \\ \end{gathered}$

Form the union those five sets of pairs. You will have a set of 22 pairs.
That set is the equivalence relation that you have been asked for.

5. So R={(1,1),(2,2),(3,3),(4,4),(5,5),(6,6),(7,7),(8,8) ,(9,9),(10,10),(1,2),(2,1),(3,4),(4,3),(4,5),(5,4) ,(5,3),(3,5) ,(6,7),(7,6),(8,10),(10,8)}

yes?

6. Originally Posted by Kitizhi
So R={(1,1),(2,2),(3,3),(4,4),(5,5),(6,6),(7,7),(8,8) ,(9,9),(10,10),(1,2),(2,1),(3,4),(4,3),(4,5),(5,4) ,(5,3),(3,5) ,(6,7),(7,6),(8,10),(10,8)}
YES!

7. ## Help with equivalence relation of a particition

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

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

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

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

Am I doing this correctly? On the right track?