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Closures 211
(b) Show that if R is a strict total order, then S is a total order.
∗
5. Suppose R is a relation on A. Let S = R \ i A .
(a) Prove that S is the largest element of the set {T ⊆ A × A | T ⊆ R
and T is irreflexive}.
(b) Prove that if R is a partial order on A, then S is a strict partial order
on A.
6. Let P be the set of all people, and let R ={(p, q) ∈ P × P | the person
p is a parent of the person q}.
(a) Let S be the transitive closure of R. Describe the relation S.
−1
(b) Describe the relation S ◦ S .
∗
7. Suppose R is a relation on A.
(a) Prove that R is reflexive iff R is its own reflexive closure.
(b) Do similar theorems hold for symmetry and transitivity? Justify your
answers with proofs or counterexamples.
8. Suppose R is a relation on A, and let S be the symmetric closure of R.
Prove that Dom(S) = Ran(S) = Dom(R) ∪ Ran(R).
∗
9. Suppose R is a relation on A, and let S be the transitive closure of R.
Prove that Dom(S) = Dom(R) and Ran(S) = Ran(R).
10. Suppose R is a relation on A. Let F ={T ⊆ A × A | R ⊆ T and T is
symmetric}. Complete the alternative proof of Theorem 4.5.5 suggested
in the text as follows:
(a) Prove that F = ∅.
(b) Let S =∩F. Prove that S is the symmetric closure of R.
11. Suppose R 1 and R 2 are relations on A and R 1 ⊆ R 2 .
(a) Let S 1 and S 2 be the reflexive closures of R 1 and R 2 respectively.
Prove that S 1 ⊆ S 2 .
(b) Do similar theorems hold for the symmetric and transitive closures?
Justify your answers with proofs or counterexamples.
∗ 12. Suppose R 1 and R 2 are relations on A, and let R = R 1 ∪ R 2 .
(a) Let S 1 , S 2 , and S be the reflexive closures of R 1 , R 2 , and R respec-
tively. Prove that S 1 ∪ S 2 = S.
(b) Let S 1 , S 2 , and S be the symmetric closures of R 1 , R 2 , and R respec-
tively. Prove that S 1 ∪ S 2 = S.
(c) Let S 1 , S 2 , and S be the transitive closures of R 1 , R 2 , and R respec-
tively. Prove that S 1 ∪ S 2 ⊆ S, and give an example to show that it
may happen that S 1 ∪ S 2 = S.
13. Suppose R 1 and R 2 are relations on A, and let R = R 1 ∩ R 2 .
(a) Let S 1 , S 2 , and S be the reflexive 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.