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Mathematics LibreTexts

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 \(A \subseteq C\).