Prove that the transitive closure of R is indeed R + := R ∪ R 2 ∪ R 3 ∪ · · ·, as follows: show that if S ⊇ R is any transitive relation, then R k ⊆ S. (We’d also need to prove that R + is transitive,...

Prove that the transitive closure of R is indeed R + := R ∪ R
²
∪ R
³
∪ · · ·, as follows: show that if S ⊇ R is any transitive relation, then R k ⊆ S. (We’d also need to prove that R + is transitive, but you can omit this part of the proof. You may find a recursive definition of R k most helpful: R 1 = R and R k = R ◦ R
^k−1
.)