## 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.