0
$\begingroup$

Let $R_1$ and $R_2$ be equivalence relations on a set $S$. Their intersection is on the relations considered as sets of ordered pairs. Identify the equivalence classes of the relation $R_1 \cap R_2$.

I don't know how to go about showing this with proof, any help would be appreciated!

$\endgroup$
1
  • 2
    $\begingroup$ $[x]_{R_1\cap R_2}=\{y\in S\mid y(R_1\cap R_2)x\}$. Work this out. $\endgroup$ – drhab Apr 13 '16 at 16:59
1
$\begingroup$

If $E$ denotes an equivalence relation then the following statements are equivalent:

  • $\langle x,y\rangle\in E$
  • $y\in[x]_E$

Here $[x]_E$ denotes the equivalence class represented by $x$.

By definition of intersection:$$\langle x,y\rangle\in R_1\cap R_2\iff\langle x,y\rangle\in R_1\wedge\langle x,y\rangle\in R_2$$ Apparantly this statement can be translated into:$$y\in[x]_{R_1\cap R_2}\iff y\in[x]_{R_1}\wedge y\in[x]_{R_2}$$

Based on this find a relation between $[x]_{R_1\cap R_2}$, $[x]_{R_1}$ and $[x]_{R_2}$.

$\endgroup$
0
$\begingroup$

Maybe this will help:

equivalence relation

This is a visual representation of two equivalence relations. Do you think you can translate what is going on here into the language of math?

$\endgroup$
1
  • $\begingroup$ Did you paint this guy? $\endgroup$ – B. Pasternak Apr 13 '16 at 17:19

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.