Problems

1. True or False

   Tell whether the following statements are true or false.

   1. ℚ × ℚ ⊆ ℝ × ℝ
   2. aRb ⇒ (a, b) ∈ R
   3. aRb ⇒ bRa
   4. aRa
   5. For any two sets A, B, A × B = B × A
   6. For some sets A, B, A × B = B × A
2. Relations

   For the set A = {1, 2, 3, 4}, write out the ordered pairs in the relation R on A if

   1. xRy ⇔ x < y
   2. xRy ⇔ x = y
   3. xRy ⇔ x divides y
   4. xRy ⇔ x is a multiple of y
3. Four Basic Cartesian Products

   Given A = {1, 2, 3}, B = {a, b}. Find the following Cartesian products.

   1. A × B
   2. B × A
   3. A × A
   4. B × B
4. Cartesian Products

   For each of the following pair of sets A and B, find the Cartesian products A × B and B × A.

   1. A = {0, 2}, B = {−1, 0}
   2. A = {a, b}, B = {b, c}
   3. A = ℝ, B = ℕ
   4. A = ℤ, B = ℕ
   5. A = ℝ, B = {−1, 0, 1}
5. Graphing a Relation

   Draw a sketch of the following relations.

   1. R = {(x, y) ∈ ℝ × ℝ : x² + y² = 1}
   2. R = {(x, y) ∈ ℝ × ℝ : y = sin x}
   3. R = {(x, y) ∈ ℝ × ℝ : x = y²}
   4. R = {(x, y) ∈ ℝ × ℝ : |x| ≤ 1, |y| ≥ 1}
   5. R = {(x, y) ∈ ℕ × ℕ : x ≡ 0(mod3), y ≡ 1(mod3)}
   6. R = {(x, y) ∈ ℕ × ℕ : x divides y}
6. Algebra of Relations

   Given are the closed intervals

   

   on the real line ℝ. Sketch the following relations in the plane.

   1. R = (A ∪ B) × C ⊆ ℝ × ℝ
   2. R = (A ∩ B) × C ⊆ ℝ × ℝ
   3. R = (A × B) ∪ (A × C) ⊆ ℝ × ℝ
   4. R = A × (A ∪ C) ⊆ ℝ × ℝ
7. Naming a Relation

   Give common names that describe the following relations on . Then find the inverse relation. What is a name for the relation and inverse relation?

   1. R = {(1, 1), (2, 2), (3, 3)}
   2. R = {(1, 2), (1, 3), (2, 3)}
   3. R = {(1, 1), (1, 2), (1, 3), (2, 2), (2, 3), (3, 3)}
8. Important Types of Relations

   Two important types of relations are injections (1–1) and surjections (onto), whose definitions we have seen in Section 2.4 in conjunction with the cardinality of sets. The definition of an injection and surjection defined especially for relations are as follows:

   • Surjective Relation: A relation R ⊆ X × Y is surjective if
     
   • Injective Relation: A relation R ⊆ X × Y is injective if
     

   For X = Y = {1, 2, 3}, give an example of an injective and surjective relation from X to Y.

9. Meaning of Relations

   For each of the following, describe the members of the relation R.

   1. 
   2. R ⊆ ℚ × ℚ, R = Ø
   3. R ⊆ ℝ × ℝ, R = ℝ × ℝ
10. Inverse Relation of Compositions

    Given A = {1, 2, 3} verify the following identity for the inverse of a composition:

    

    for relations

    
11. Composition of Relations

    Find the composition of the relations S ∘ R given the sets

    

    and the relations

    
12. Cartesian Product Identities

    Prove the identity

    
13. Number of Relations

    If a set A has m elements and B has n elements, show that the number of relations from A to B is 2^mn.

14. Graphing Relations and Their Inverses

    Graph the following relations and their inverses.

    1. R ⊆ ℝ × ℝ, xRy ⇔ y = 1/x
    2. R ⊆ ℝ × ℝ, xRy ⇔ y = e^x
    3. R ⊆ ℝ × ℝ, R = {(x, y) : |x| + |y| = 1}
15. Counting Relations I

    What is the total number of relations that can be defined on the set A = {1, 2}?

16. Counting Relations II

    What is the total number of relations that can be defined on the set A = {1, 2, 3}?

17. Converse of a Binary Relation

    If R is a binary relation on a set A, then the converse relation on A is defined by the relation . State in English or write out in ordered pairs the converse of the following relations R.

    1. The relation “is the mother of.”
    2. The relation “is a uncle of.”
    3. The relation “<” on the real numbers.
    4. The relation “=” on the real numbers.
18. Blood Typing

    There are four blood, types A, B, AB, and O.

    • Type A can receive blood from type A and O.
    • Type B can receive blood from type B and O.
    • Type AB can receive blood from all types.
    • Type O can receive blood from only type O.

    Given the set S = {A, B, AB, O}, define a binary relation R on S as follows:

    
    1. Write out the ordered pairs of R.
    2. Draw a directed graph of the relation R.
    3. Define the converse of R by . What is the interpretation of the converse?
19. Internet Research

    There is a wealth of information related to topics introduced in this section just waiting for curious minds. Try aiming your favorite search engine toward important relations in mathematics, visualizing mathematical relations, and composition of relations.