Union, Intersection, Difference. Complements. Venn Diagrams

The following is the list of problems for Sections 1.5, 1.6 and 1.7 of the Book of Proof (pages 18, 20, 23). There is a forum open at the end, so you can ask questions. It is a great way to interact with the instructor and with other students in your class, should you need some assistance with any question. Please, do not post solutions.

Exercises for Section 1.5

1. Suppose \( A = \{ 4,3,6, 7,1,9 \}, B = \{ 5,6,8,4 \} \) and \( C = \{ 5, 8, 4 \} \). Find:

   1. \( A \cup B \)
   2. \( A \cap B \)
   3. \( A \setminus B \)
   4. \( A \setminus C \)
   5. \( B \setminus A \)
   6. \( A \cap C \)
   7. \( B \cap C \)
   8. \( B \cup C \)
   9. \( C \setminus B \)

2. Suppose \( A = \{ 0, 2, 4, 6, 8 \}, B = \{ 1, 3, 5, 7 \} \) and \( C = \{ 2, 8, 4 \} \). Find:

   1. \( A \cup B \)
   2. \( A \cap B \)
   3. \( A \setminus B \)
   4. \( A \setminus C \)
   5. \( B \setminus A \)
   6. \( A \cap C \)
   7. \( B \cap C \)
   8. \( B \cup C \)
   9. \( C \setminus B \)

3. Suppose \( A = \{ 0, 1 \} \) and \( B = \{ 1, 2 \} \). Find:

   1. \( ( A \times B) \cap ( B \times B) \)
   2. \( ( A \times B) \cup ( B \times B) \)
   3. \( ( A \times B) \setminus ( B \times B) \)
   4. \( (A \cap B) \times A \)
   5. \( (A \times B) \cap B \)
   6. \( \mathscr{P}(A) \cap \mathscr{P}(B) \)
   7. \( \mathscr{P}(A) \setminus \mathscr{P}(B) \)
   8. \( \mathscr{P}(A \cap B) \)
   9. \( \mathscr{P}(A \times B) \)

4. Suppose \( A = \{ b, c, d \} \) and \( B = \{ a, b \} \). Find:

   1. \( ( A \times B) \cap ( B \times B) \)
   2. \( ( A \times B) \cup ( B \times B) \)
   3. \( ( A \times B) \setminus ( B \times B) \)
   4. \( (A \cap B) \times A \)
   5. \( (A \times B) \cap B \)
   6. \( \mathscr{P}(A) \cap \mathscr{P}(B) \)
   7. \( \mathscr{P}(A) \setminus \mathscr{P}(B) \)
   8. \( \mathscr{P}(A \cap B) \)
   9. \( \mathscr{P}(A) \times \mathscr{P}(B) \)
5. Sketch the sets \( X = [1,3] \times [1,3] \) and \( Y = [2,4] \times [2,4] \) on the plane \( \mathbb{R}^2 \). On separate drawings, shade in the sets \( X \cup Y, X \cap Y, X \setminus Y \) and \( Y \setminus X \).
6. Sketch the sets \( X= [-1,3]\times [0,2] \) and \( Y = [0,3] \times [1,4] \) on the plane \( \mathbb{R}^2 \). On separate drawings, shade in the sets \( X \cup Y, X \cap Y, X \setminus Y \) and \( Y \setminus X \).
7. Sketch the sets \( X = \{(x,y) \in \mathbb{R}^2 : x^2 + y^2 \leq 1 \} \) and \( Y = \{ (x,y) \in \mathbb{R}^2 : x \geq 0 \} \) on \( \mathbb{R}^2 \). On separate drawings, shade in the sets \( X \cup Y, X \cap Y, X \setminus Y \) and \( Y \setminus X \).
8. Sketch the sets \( X = \{(x,y) \in \mathbb{R}^2 : x^2 + y^2 \leq 1 \} \) and \( Y = \{ (x,y) \in \mathbb{R}^2 : -1 \leq y \leq 0 \} \) on \( \mathbb{R}^2 \). On separate drawings, shade in the sets \( X \cup Y, X \cap Y, X \setminus Y \) and \( Y \setminus X \).
9. Is the statement \( (\mathbb{R} \times \mathbb{Z}) \cap (\mathbb{Z} \times \mathbb{R} ) = \mathbb{Z} \times \mathbb{Z} \) true or false? What about the statement \( (\mathbb{R} \times \mathbb{Z}) \cup (\mathbb{Z} \times \mathbb{R} ) = \mathbb{R} \times \mathbb{R} \)?
10. Do you think the statement \( ( \mathbb{R} \setminus \mathbb{Z} ) \times \mathbb{N} = (\mathbb{R} \times \mathbb{N} ) \setminus (\mathbb{Z} \times \mathbb{N} ) \) is true, or false? Justify.

Exercises for Section 1.6

1. Let \( A = \{ 4,3, 6,7,1,9 \} \) and \( B = \{ 5,6,8,4 \} \) have universal set \( U = \{ 0, 1, 2, \dotsc, 10\} \). Find:

   1. \( A^\complement \)
   2. \( B^\complement \)
   3. \( A \cap A^\complement \)
   4. \( A \cup A^\complement \)
   5. \( A \setminus A^\complement \)
   6. \( A \setminus B^\complement \)
   7. \( A^\complement \setminus B^\complement \)
   8. \( A^\complement \cap B \)
   9. \( (A^\complement \cap B)^\complement \)

2. Let \( A = \{ 0, 2, 4, 6, 8 \} \) and \( B = \{ 1, 3, 5, 7 \} \) have universal set \( U = \{ 0, 1, 2, \dotsc, 8 \} \). Find:

   1. \( A^\complement \)
   2. \( B^\complement \)
   3. \( A \cap A^\complement \)
   4. \( A \cup A^\complement \)
   5. \( A \setminus A^\complement \)
   6. \( (A \cap B)^\complement \)
   7. \( A^\complement \cap B^\complement \)
   8. \( (A \cap B)^\complement \)
   9. \( A^\complement \times B \)
3. Sketch the set \( X = [1,3] \times [1,2] \) on the plane \( \mathbb{R}^2 \). On separate drawings, shade in the sets \( X^\complement \) and \( X^\complement \cap \big( [0,2] \times [0,3] \big) \).
4. Sketch the set \( X = [-1,3] \times [0,2] \) on the plane \( \mathbb{R}^2 \). On separate drawings, shade in the sets \( X^\complement \) and \( X^\complement \cap \big( [-2,4] \times [-1,3] \big) \).
5. Sketch the set \( X = \{ (x,y) \in \mathbb{R}^2 : 1 \leq x^2 + y^2 \leq 4 \} \) on the plane \( \mathbb{R}^2 \). On a separate drawing, shade in the set \( X^\complement \).
6. Sketch the set \( X = \{ (x,y) \in \mathbb{R}^2 : y < x^2 \} \) on \( \mathbb{R}^2 \). Shade in the set \( X^\complement \).

Exercises for Section 1.7

1. Draw a Venn diagram for \( A^\complement \).
2. Draw a Venn diagram for \( B \setminus A \).
3. Draw a Venn diagram for \( (A\setminus B) \cap C \).
4. Draw a Venn diagram for \( (A \cup B) \setminus C \).
5. Draw Venn diagrams for \( A \cup (B \cap C) \) and \( A \cup B) \cap (A \cup C) \). Based on your drawings, do you think \( A \cup (B \cap C) = A \cup B) \cap (A \cup C) \)?
6. Draw Venn diagrams for \( A \cap (B \cup C) \) and \( A \cap B) \cup (A \cap C) \). Based on your drawings, do you think \( A \cap (B \cup C) = A \cap B) \cup (A \cap C) \)?
7. Suppose set \( A \) and \( B \) are in a universal set \( U \). Draw Venn diagrams for \( (A \cap B)^\complement \) and \( A^\complement \cup B^\complement \). Based on your drawings, do you think it is true that \( (A \cap B)^\complement = A^\complement \cup B^\complement \)?
8. Suppose set \( A \) and \( B \) are in a universal set \( U \). Draw Venn diagrams for \( (A \cup B)^\complement \) and \( A^\complement \cap B^\complement \). Based on your drawings, do you think it is true that \( (A \cup B)^\complement = A^\complement \cap B^\complement \)?
9. Draw a Venn diagram for \( (A \cap B) \setminus C \).
10. Draw a Venn diagram for \( (A \setminus B ) \cup C \).

Following are Venn diagrams for expressions involving sets \( A, B \) and \( C \). Write the corresponding expression.

12.

13.

14.