Is $(\prod R_i)\circ (\prod R_i)$ the same as $\prod R_i\circ R_i$?

Question

For each $i\in I$, $R_i$ is a (binary) relation (on a set $X_i$). Is $(\prod R_i)\circ (\prod R_i)$ the same as $\prod R_i\circ R_i$ as relations on $\prod_{i\in I}X_i$?

Answer 1

This is quite easy to see from the definitions, and I think you'll be better off putting it together yourself than having a full answer provided. I expect you know the definitions involved, but just in case you're unclear on anything:

Two elements $(x_i)_{i\in I}$ and $(y_i)_{i\in I}$ of $\prod_{i\in I} X_i$ are related by $\prod_{i\in I} \rho_i$, where each $\rho_i$ is a relation on $X_i$, if $x_i\, \rho_i\, y_i$ for all $i\in I$.

If $\rho$ and $\sigma$ are relations on a set $X$, then $x,y\in X$ are related by $\rho\circ \sigma$ if there is some $z\in X$ such that $x\,\rho\, z\,\sigma\, y$.

Now mix the two definitions together in two different ways, and check whether you get the same thing both times.