Question

Are transitive relations closed under intersaction

Answer 1

Answer:

In mathematics, the transitive closure of a binary relation R on a set X is the smallest relation on X that contains R and is transitive. ... If the binary relation itself is transitive, then the transitive closure is that same binary relation; otherwise, the transitive closure is a different relation.

Answer 2

Answer:

Then since R1 and R2 are both transitive, we have (x, z) ∈ R1 and (x, z) ∈ R2. So (x, z) ∈ R1 ∩ R2 and the intersection is transitive. This shows that R1 ∩R2 is an equivalence relation; that is, equivalence relations on a set A are closed under intersection.