There's a question I dont understand.
If R and S are equivalence relations on a set , then which of the following is guaranteed to be an equivalence relation again:
- R ∪ S
- R ∩ S
- R * S
- R / S
The answer given is R ∩ S, but to my understanding I thought it was impossible to intersect two equivalent sets. So I assume that the only possible answers are all but R ∩ S
The question does not ask about intersecting equivalence sets, it asks about intersecting the equivalent relations themselves. For example on the set $A:= \{ 1,2,3,4\}$ I can define the equivalence relation $R$ on $A$ to mean that $x,y \in A$ are equivalent iff $x$ and $y$ have the same parity (in the sense of either both being even or both being odd. The equivalence sets you talk about are $\{1,3 \}$ and $\{2,4\}$ in my example, since those are the sets of numbers having the same parity. However the equivalence relation $R$ is the set of all tuples over $A$ satisfying the condition. So we get $R = \{(1,1),(2,2),(3,3),(4,4),(1,3),(3,1),(2,4),(4,2) \}$.
The question talks about intersecting two such sets $R$ and $S$, and it is straightforward checking that $R \cap S$ satisfies the definition of an equivalence relation.