11 truct examples of equivalence relations R and Re such tha

11. truct examples of equivalence relations R, and Re such that Ri U is not tran sitive. (See Fact 3.45.)

Solution

Let A={1,2,3},

R_1={(1,1),(2,2),(3,3),(1,2),(2,1)} and R_2={(1,1),(2,2),(3,3),(2,3),(3,2)}

We can check that R_1 and R_2 are equivalence relation.

Also,

R_1 R_2 is reflexive and symmetric but not transitive because

R_1 R_2 contains both (1,2) and (2,3) but not (1,3).

so, the two equivalence relations are

R_1={(1,1),(2,2),(3,3),(1,2),(2,1)} and R_2={(1,1),(2,2),(3,3),(2,3),(3,2)}

such that R_1 R_2 is not transitive.