# equivalence relations

• Nov 2nd 2008, 05:59 PM
junebug5389
equivalence relations
So this is a problem for one of my classes. I don't even know where to start...thanks for any help!

Let A be a non-empty set, and let E1 and E2 be equivalence relations on A with associated partitions D1 and D2 , respectively. Let E be the equivalence relation on A defined by E = E1 ∩ E2 , and D its associated partition. Come up with a proposition describing the relationship between
D1 , D2 and D. Prove your result.
• Nov 3rd 2008, 03:15 AM
Plato
Quote:

Originally Posted by junebug5389
Let A be a non-empty set, and let E1 and E2 be equivalence relations on A with associated partitions D1 and D2 , respectively. Let E be the equivalence relation on A defined by E = E1 ∩ E2 , and D its associated partition. Come up with a proposition describing the relationship between D1 , D2 and D. Prove your result.

Given an equivalence relation $\displaystyle E$ on a set $\displaystyle A$, each cell in a partition determined by $\displaystyle E$ is one of these sets: $\displaystyle E_x = \left\{ {y \in A:\left( {x,y} \right) \in E} \right\}$.
Therefore the partition determined by $\displaystyle D$ is $\displaystyle \left\{ {E1_x \cap E2_x :x \in A} \right\}$.