1. ## Equivalence relations

I am having a hard time with equivalence relations. This problem was set before me and I was wondering if anyone could point me in the right direction of how to work it out. Thanks!

a) Let R1 be an equivalence relation on S and R2 be an equivalence relation on T. Let R3 be a relation on the Cartesian product of S and T given by (a,r)R3 (b,s) iff aR1b and rR2s. Prove that R3 is an equivalence relation.
b) If R1 has n1 classes and R2 has n2 classes, how many does R3 have? (my guess is n3)
c) Suppose that S intersect T = { }. Explain how to construct an R4 on S U T in a very short explanation.

2. Originally Posted by arizona11
a) Let R1 be an equivalence relation on S and R2 be an equivalence relation on T. Let R3 be a relation on the Cartesian product of S and T given by (a,r)R3 (b,s) iff aR1b and rR2s. Prove that R3 is an equivalence relation.
b) If R1 has n1 classes and R2 has n2 classes, how many does R3 have? (my guess is n3)
c) Suppose that S intersect T = { }. Explain how to construct an R4 on S U T in a very short explanation.
What difficulties are you having with part a)? It really is straightforward.

For part b) $\displaystyle n_1\times n_2$.
Each equivalence class from $\displaystyle R_1$ can be paired with every equivalence class from $\displaystyle R_2$ to give one equivalence class in $\displaystyle R_3$.

For part c) define $\displaystyle R_4$ on $\displaystyle S \cup T$ by $\displaystyle xR_4y$
If and only if $\displaystyle \left\{ {x,y} \right\} \subseteq S \vee \left\{ {x,y} \right\} \subseteq T$ then $\displaystyle xR_4y=xR_1y~\vee ~ xR_4y=xR_2y$.

3. I wasn't taught how to do it. It was just thrown at us. Thanks for the help!