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212 Relations
(b) Let S 1 , S 2 , and S be the symmetric closures of R 1 , R 2 , and R respec-
tively. What is the relationship between S 1 ∩ S 2 and S? Justify your
conclusions with proofs or counterexamples.
(c) Let S 1 , S 2 , and S be the transitive closures of R 1 , R 2 , and R respec-
tively. What is the relationship between S 1 ∩ S 2 and S? Justify your
conclusions with proofs or counterexamples.
14. Find an example of relations R 1 and R 2 on some set A such that, if we
let R = R 1 \ R 2 and we let S 1 , S 2 , and S be the transitive closures of R 1 ,
R 2 , and R respectively, then S 1 \ S 2 ⊆ S and S ⊆ S 1 \ S 2 .
∗ 15. Suppose R is a relation on A. The reflexive symmetric closure of R is
the smallest set S ⊆ A × A such that R ⊆ S, S is reflexive, and S is
symmetric, if there is such a smallest set. Prove that every relation has a
reflexive symmetric closure.
16. Suppose R is a relation on A, and let S be the reflexive closure of R.
(a) Prove that if R is symmetric, then so is S.
(b) Prove that if R is transitive, then so is S.
17. Suppose R is a relation on A, and let S be the transitive closure of R. Prove
that if R is symmetric, then so is S. (Hint: Assume that R is symmetric.
Prove that R ⊆ S −1 and S −1 is transitive. What can you conclude about
−1
S and S ?)
18. Suppose R is a relation on A. The symmetric transitive closure of R is
∗
the smallest set S ⊆ A × A such that R ⊆ S, S is symmetric, and S is
transitive, if there is such a smallest set.
Let Q be the symmetric closure of R, and let S be the transitive closure
of Q.Also,let Q bethetransitiveclosureof R,andlet S bethesymmetric

closure of Q .

(a) Prove that S is the symmetric transitive closure of R. (Hint: Use
exercise 17.)
(b) Prove that S ⊆ S.

(c) Must it be the case that S = S? Justify your answer with either a
proof or a counterexample.
19. Consider the following putative theorem:
Theorem? Suppose R is a reflexive, antisymmetric relation on A. Let S
be the transitive closure of R. Then S is a partial order on A.

Is the following proof correct? If so, what proof strategies does it use? If
not, can it be fixed? Is the theorem correct?

Proof. R is already reflexive and antisymmetric. To form the relation
S we add more ordered pairs to make it transitive as well. Thus, S is
reflexive, antisymmetric, and transitive, so it is a partial order.