# Determine whether each of these combinations of R 1 and R 2 must be an equivalence relation.

I have this question but not really sure how to do it when there is union and interception symbol. I am easily confuse when this 2 symbol appear. From my understanding I know that equivalence relation means that it must be reflective, symmetric and transitive. Can anyone guide me. Thanks!

Suppose that $R_1$ and $R_2$ are equivalence relations on the set S. Determine whether each of these combinations of R 1 and R 2 must be an equivalence relation. a) $R_1 \cup R_2$ b) $R_1 \cap R_2$

## 1 Answer

For union, transitivity fails.

For instance $aR_1 b$, $bR_2 c$ does not imply either of $aR_1 c$ or $aR_2 c$.

Intersection is true, denote $R_1\cap R_2$ by $\sim$.

Reflexive: $aR_1a$, $a R_2 a$, so $a\sim a$.

Symmetric: Given $a \sim b$, $a R_1 b$ and $a R_2 b$.

Thus $bR_1a$ and $bR_2a$, i.e. $b\sim a$.

Similarly for transivity, one can show that $a\sim b$ and $b\sim c$ implies $a\sim c$.

• So when I encounter this type of question, I should write down some variable and try to see if it can fulfill the relations? For the interception, may I have more details? I am not very sure how to show it. Thanks! – user292965 Mar 29 '16 at 13:28
• I wrote some details on intersection. Hope it helps – yoyostein Mar 29 '16 at 14:20