I have to prove that this is an equivalence relation: $B R C := A \cap B = A \cap C$
I do unterstand that I would then have to prove that this is reflexive, symmetrical and transitive, but I don't know how. Can anyone help me, I would be very grateful,
Thank you
Reflexive: $BRB$ because $A \cap B= A \cap B$ this is trivial (think which elements are in right, are they in the left also? What about the other possibility?
Symmetric:
$BRC$ should imply$CRB$
This is true because $A \cap B= A \cap C$ and this implies the other way ($A \cap C= A \cap B$ (nothing has changed, think about how switching orders of intersections doesn’t change anything)
Transitive:
$BRC$ and $CRD$ should imply $BRD$
Well, if intersection $A$ and $B$ is equal to intersection $A$ and $C$ and intersection $A$ and $D$ is equal to intersection $A$ and $C$, this could mean only one thing (what does it mean)?
Please show some work if you need help/ point out where exactly you need help with.