I have this question and would need help on how to find $R_1 \cup R_2$. My working for $R_1 \cap R_2$ is shown below:
Let $R_1$ and $R_2$ be the “congruent modulo 3” and the “congruent modulo 4” relations on the set of integers respectively. That is, $R_1 = \{(a,b)\ |\ a \equiv b(mod\ 3)\}$ and $R_2 = \{(a,b)\ |\ a\equiv b(mod\ 4)\}.$ Find $R_1\cap R_2$ and $R_1 \cup R_2.$
For $(a,b) \in R_1 \cap R_2$:
$a - b = 3r$ for some $ r \in \Bbb{Z}$ and
$a - b = 4t, $ for some $t \in \Bbb{Z}.$
Therefore, I can conclude that $a - b = 12k$ for some $k \in \Bbb{Z}.$
Hence, $R_1 \cap R_2$ implies that $a \equiv b(mod\ 12).$