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# 8.5: Exercises

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Use the methods introduced in this chapter to prove the following statements.

Exercise $$\PageIndex{1}$$

Prove that $$\{12n : n \in \mathbb{Z}\} \subseteq \{2n : n \in \mathbb{Z}\} \cap \{3n : n \in \mathbb{Z}\}$$.

Exercise $$\PageIndex{2}$$

Prove that $$\{6n : n \in \mathbb{Z}\} \subseteq \{2n : n \in \mathbb{Z}\} \cap \{3n : n \in \mathbb{Z}\}$$.

Exercise $$\PageIndex{3}$$

If $$k \in \mathbb{Z}$$, then $$\{n \in \mathbb{Z} : n|k\} \subseteq \{n \in \mathbb{Z} : n|k^2\}$$.

Exercise $$\PageIndex{4}$$

If $$m, n \in \mathbb{Z}$$, then $$\{x \in \mathbb{Z} : mn|x\} \subseteq \{x \in \mathbb{Z} : m|x\} \cap \{x \in \mathbb{Z} : n|x\}$$.

Exercise $$\PageIndex{5}$$

If p and q are positive integers, then $$\{pn : n \in \mathbb{N}\} \cap \{qn : n \in \mathbb{N}\} = \emptyset\}$$.

Exercise $$\PageIndex{6}$$

Suppose A, B and C are sets. Prove that if $$A \subseteq B$$, then $$A-C \subseteq B-C$$.

Exercise $$\PageIndex{7}$$

Suppose A, B and C are sets. If $$B \subseteq C$$, then $$A \times B \subseteq A \times C$$.

Exercise $$\PageIndex{8}$$

If A, B and C are sets, then $$A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$$.

Exercise $$\PageIndex{9}$$

If A,B and C are sets, then $$A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$$.

Exercise $$\PageIndex{10}$$

If A and B are sets in a universal set U, then $$\overline{A \cap B} = \overline{A} \cup \overline{B}$$.

Exercise $$\PageIndex{11}$$

If A and B are sets in a universal set U, then $$\overline{A \cup B} = \overline{A} \cap \overline{B}$$.

Exercise $$\PageIndex{12}$$

If A, B and C are sets, then $$A-(B \cap C) = (A-B) \cup (A-C)$$.

Exercise $$\PageIndex{13}$$

If A, B and C are sets, then $$A-(B \cup C) = (A-B) \cap (A-C)$$.

Exercise $$\PageIndex{14}$$

If A, B and C are sets, then $$(A \cup B)-C = (A-C) \cup (B-C)$$.

Exercise $$\PageIndex{15}$$

If A, B and C are sets, then $$(A \cap B)-C = (A-C) \cap (B-C)$$.

Exercise $$\PageIndex{16}$$

If A, B and C are sets, then $$A \times (B \cup C) = (A \times B) \cup (A \times C)$$.

Exercise $$\PageIndex{17}$$

If A, B and C are sets, then $$A \times (B \cap C) = (A \times B) \cap (A \times C)$$.

Exercise $$\PageIndex{18}$$

If A, B and C are sets, then $$A \times (B - C) = (A \times B) - (A \times C)$$.

Exercise $$\PageIndex{19}$$

Prove that $$\{9^n : n \in \mathbb{Z}\} \subseteq \{3^n : n \in \mathbb{Z}\}$$, but $$\{9^n : n \in \mathbb{Z} \ne \{3^n : n \in \mathbb{Z}\}$$.

Exercise $$\PageIndex{20}$$

Prove that $$\{9^n : n \in \mathbb{Q}\} = \{3^n : n \in \mathbb{Q}\}$$

Exercise $$\PageIndex{21}$$

Suppose A and B are sets. Prove $$A \subseteq B$$ if and only if $$A-B = \emptyset$$.

Exercise $$\PageIndex{22}$$

Let A and B be sets. Prove that $$A \subseteq B$$ if and only if $$A \cap B = A$$.

Exercise $$\PageIndex{23}$$

For each $$a \in \mathbb{R}$$, let $$A_{a} = \{(x, a(x^2-1)) \in \mathbb{R}^2 : x \in \mathbb{R}\}$$. Prove that $$\bigcap_{a \in \mathbb{R}} A_{a} = \{(-1, 0), (1, 0)\}$$.

Exercise $$\PageIndex{24}$$

Prove that $$\bigcap_{x \in \mathbb{R} [3-x^2, 5+x^2] = [3, 5]$$.

Exercise $$\PageIndex{25}$$

Suppose A, B, C and D are sets. Prove that $$(A \times B) \cup (C \times D) \subseteq (A \cup C) \times (B \cup D)$$.

Exercise $$\PageIndex{26}$$

Prove that $$\{4k+5 : k \in \mathbb{Z} = \{4k+1 : k \in \mathbb{Z}\}$$.

Exercise $$\PageIndex{27}$$

Prove that $$\{12a+4b : a, b \in \mathbb{Z} = \{4c : c \in \mathbb{Z}\}$$.

Exercise $$\PageIndex{28}$$

Prove that $$\{12a+25b : a, b \in \mathbb{Z} = \mathbb{Z}\}$$.

Exercise $$\PageIndex{29}$$

Suppose $$A \ne \emptyset$$. Prove that $$A \times B \subseteq A \times C$$ if and only if $$B \subseteq C$$.

Exercise $$\PageIndex{30}$$

Prove that $$(\mathbb{Z} \times \mathbb{N}) \cap (\mathbb{N} \times \mathbb{Z}) = \mathbb{N} \times \mathbb{N}$$.

Exercise $$\PageIndex{31}$$

Suppose $$B \ne \emptyset$$ and $$A \times B \subseteq B \times C$$. Prove that $$A \subseteq C$$.