Question

Consider the empty set as a relation, R, on any non-empty set S. Prove or disprove: R is transitive.

Answer 1

Let's prove with an example, let's consider a statement, if A then B.
This is false only when A is true and B is false.

To prove that , let's consider : "if x is a human then x can eat" is not always going to be true, we need to give an example of something that is human but cannot eat.

Relation R is transitive for all x,y,z

if (x,y)∈R
and
(y,z)∈R
then (x,z)∈R.

If R is an empty relation, then (x,y)∈R and (y,z)∈R cannot be true.
But the whole statement cannot be false.
This proves that R is transitive on any non-empty set S.