a problem with relations

• Apr 6th 2009, 10:48 AM
stewie griffin
a problem with relations
Let M be the set of all the relation on A={1,2,3}
Let S be the following relation on M:
for R1, R2 relations on A, (R1,R2) Î S iff R1oR2 = R2oR1
(o - composition)
prove that S is not an equivalence relation on M

I am having a little problem with this...
I guess I need to prove that S is not transitive but I can't figure it out....

any directions would be appreciated
thanks

• Apr 6th 2009, 11:45 AM
Plato
Let $\displaystyle R_1 = \left\{ {(1,1),(2,2),(3,3)} \right\}\;,\,R_2 = \left\{ {(1,2),(2,3)} \right\},\;\& \,R_3 = \left\{ {(2,1),(3,2)} \right\}$

Are these true? $\displaystyle R_1 \circ R_2 = R_2 \circ R_1 ~~\& ~~R_1 \circ R_3 = R_3 \circ R_1$

Is this true? $\displaystyle R_3 \left( \mathcal{S} \right)R_2$
• Apr 6th 2009, 12:12 PM
stewie griffin
Quote:

Originally Posted by Plato
Let $\displaystyle R_1 = \left\{ {(1,1),(2,2),(3,3)} \right\}\;,\,R_2 = \left\{ {(1,2),(2,3)} \right\},\;\& \,R_3 = \left\{ {(2,1),(3,2)} \right\}$

Are these true? $\displaystyle R_1 \circ R_2 = R_2 \circ R_1 ~~\& ~~R_1 \circ R_3 = R_3 \circ R_1$

Is this true? $\displaystyle R_3 \left( \mathcal{S} \right)R_2$

I think
$\displaystyle R_2 \circ R_3 = \left\{ {(2,2),(3,3)} \right\}$

$\displaystyle R_3 \circ R_2 = \left\{ {(2,2),(1,1)} \right\}$
• Apr 6th 2009, 12:31 PM
Plato
Quote:

Originally Posted by stewie griffin
I think
$\displaystyle R_2 \circ R_3 = \left\{ {(2,2),(3,3)} \right\}$

$\displaystyle R_3 \circ R_2 = \left\{ {(2,2),(1,1)} \right\}$

That is correct. But both are related to $\displaystyle R_1$.
What is wrong with that?
• Apr 6th 2009, 10:28 PM
stewie griffin
$\displaystyle R_3 \left( \mathcal{S} \right)R_2 \iff R_3 \circ R_2 = R_2 \circ R_3$
but $\displaystyle R_3 \circ R_2 \ne R_2 \circ R_3$