I don't fully understand how to find the equivalence classes of a relation.
Over $\mathcal P(E)$, where $E = \{1,2,3,4,5,6\}$, $ARB \iff |A\cap\{1,2\}| = |B\cap\{1,2\}|$
From what I've seen, people try to make up a formula of some sort that calculates a set with all the elements that relate to an arbitrary element.
They usually start with something like this:
Have some set $X \in \mathcal P(E)$, now consider:
$$[X] = \{Y\in \mathcal P(E) : YRX\}$$
$$= \{Y \in \mathcal P(E) : |Y \cap \{1,2\}| = |X \cap \{1,2\}|\}$$
And then they elaborate to make such formula.
I don't really get the point of that. How do you determine the equivalence classes of a relation?
Best Answer
Let $\phi(A) = |A \cap \{1,2\}|$. We see that $\phi(A) \in \{0,1,2\}$, and $ARB$ iff $\phi(A) =\phi(B)$.
So the equivalence classes are $\phi^{-1} ( \{ k \} )$ for $k=0,1,2$.