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200 Relations
(b) R ={(x, y) ∈ R × R | x ≤ y}, B ={x ∈ R | 1 ≤ x < 2}.
(c) R ={(x, y) ∈ P (N) × P (N) | x ⊆ y}, B ={x ∈ P (N) | x has at
most 5 elements}.
4. Suppose R is a relation on A. You might think that R could not be both
∗
antisymmetric and symmetric, but this isn’t true. Prove that R is both
antisymmetric and symmetric iff R ⊆ i A .
5. Suppose R is a partial order on A and B ⊆ A. Prove that R ∩ (B × B)is
a partial order on B.
6. Suppose R 1 and R 2 are partial orders on A. For each part, give either a
proof or a counterexample to justify your answer.
(a) Must R 1 ∩ R 2 be a partial order on A?
(b) Must R 1 ∪ R 2 be a partial order on A?
7. Suppose R 1 is a partial order on A 1 , R 2 is a partial order on A 2 , and
A 1 ∩ A 2 = ∅.
(a) Prove that R 1 ∪ R 2 is a partial order on A 1 ∪ A 2 .
(b) Prove that R 1 ∪ R 2 ∪ (A 1 × A 2 ) is a partial order on A 1 ∪ A 2 .
(c) Suppose that R 1 and R 2 are total orders. Are the partial orders in parts
(a) and (b) also total orders?
∗
8. Suppose R is a partial order on A and S is a partial order on B. Define a

relation T on A × B as follows: T ={((a, b), (a , b )) ∈ (A × B) × (A ×
B) | aRa and bSb }. Show that T is a partial order on A × B. If both R

and S are total orders, will T also be a total order?
9. Suppose R is a partial order on A and S is a partial order on B. Define a

relation L on A × B as follows: L ={((a, b), (a , b )) ∈ (A × B) × (A ×
B) | aRa , and if a = a then bSb }. Show that L is a partial order on

A × B. If both R and S are total orders, will L also be a total order?
10. Suppose R is a partial order on A. For each x ∈ A, let P x ={a ∈ A | aRx}.
Prove that ∀x ∈ A∀y ∈ A(xRy ↔ P x ⊆ P y ).
∗ 11. Let D be the divisibility relation defined in part 3 of Example 4.4.3. Let
B ={x ∈ Z | x > 1}. Does B have any minimal elements? If so, what are
they? Does B have a smallest element? If so, what is it?
12. Show that, as was stated in part 2 of Example 4.4.7, {X ⊆ R | X = ∅ and
∀x∀y((x ∈ X ∧ x < y) → y ∈ X)} has no minimal element.
13. Suppose R is a partial order on A. Prove that R −1 is also a partial order on
A.If R is a total order, will R −1 also be a total order?
14. Suppose R is a partial order on A, B ⊆ A, and b ∈ B. Exercise 13 shows
∗
that R −1 is also a partial order on A.
−1
(a) Prove that b is the R-largest element of B iff it is the R -smallest
element of B.
−1
(b) Prove that b is an R-maximal element of B iffitisan R -minimal
element of B.