Prove that the intersection of two equivalence relations is an equivalence relation.

Question

I am reading this chapter of the Book of Proof, and I'm stuck at the Exercise 10 of section 11.2. It is as follows.

Suppose $R$ and $S$ are two equivalence relations on a set $A$. Prove that $R \cap S$ is also an equivalence relation.

Thanks for helps!

Answer 1

Hint: Use the fact that $R$ and $S$ are EQUIVALENCE relations on THE SAME set, and hence both must be reflexive, symmetric, and transitive on that set.

Then use the definition of set intersection: $R\cap S$ is the set of all pairs of elements in the set such that $(x, y) \in R$ AND $(x, y) \in S$ or, put differently, $(x, y) \in R\cap S \iff (x, y)\in R$ and $(x, y) \in S$.

Try to figure out what elements must necessarily be in $R\cap S$ and check to see that they must then be in both $R$ and $S$.

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Another approach would be to use an indirect proof with the hints above:

"Given $R$ and $S$ are equivalence relations on a set $A$, suppose for the sake of contradiction, that $R\cap S$ is NOT an equivalence relation...". If not an equivalence relation, then $R\cap S$ fails to be reflexive and/or fails to be symmetric, and/or fails to be transitive. If you can work towards a contradiction (that this assumption must contradict the fact that both $R$ and $S$ are equivalence relations), then you are done.

Answer 2

For the solution of this exercise, you have to show that $R \cap S$ keeps the three properties of equivalence relations (reflexive, symmetric and transitive).

This means that for each $x\in R\cap S$ you have to show that $\langle x,x\rangle \in R \cap S$ and for each pair $\langle x,y\rangle \in R \cap S$, you have to show that $\langle y,x\rangle \in R \cap S$ and for each pairs $\langle x,y\rangle, \langle y,z\rangle \in R \cap S$ you have to show that $\langle x,z\rangle \in R \cap S$

Answer 3

Hint (did this in school):
Let's have 2 relations

$R- antisymmetric$

$S-antisymmetric$

I had to prove that $R \cap S$ is also antisymmetric.

$P=R \cap S$

$(x,y) \in P$

$\implies (x,y) \in R \cap S $

$\implies (x,y) \in R \wedge (x,y) \in S $

$\implies ((y,x) \notin R \wedge (y,x) \notin S) \Rightarrow(x \ne y) $

$\implies (y,x) \notin R \cap S $

$\implies (y,x) \notin P $