# Thread: Important Equivalence Relation Problem

1. ## Important Equivalence Relation Problem

Hi! I really need help on this one problem:

Suppose that R1 and R2 are equivalence relations on the set S. Determine whether each of these combinations of R1 and R2 must be an equivalence relation.

a) R1 U R2
b) R1 ^ R2 (^ = intersection)

Please respond ASAP. This assignment is due tomorrow morning. I'm really sorry for making it so last minute. Thank you so much for being super awesome!

Edit: I guess I should add that I think a) is not an equivalence relation and b) is a equivalence relation though I don't know how to prove it.

2. a) Consider $X=\{1,2,3\}$
$R_1=\{(1,1), (2,2), (1,2), (2,1),(3,3)\}$
$R_2=\{(1,1),(2,2),(3,3),(2,3),(3,2)\}$

The union fails transitivity. $(1,2),(2,3)\in R_1 \cup R_2$; however, $(1,3)\not \in R_1 \cup R_2$

b) Let $S,T$ be equivalence relations and let $R=S\cap T$.

If both S and T are equivalence relations we know $(a,a)$ is in both so it is in the intersection for all a.

If $(a,b)\in R$ then $(a,b)$ is in S and T, by symmetry $(b,a)$ is in both S and T, so it is in R.

If (a,b) and (b,c) are in R they are in both S and T, so (a,c) must be in S and T by transitivity of each separately, so (a,c) is in R as well.

3. Thanks for you help, Gamma!