Chapter 9

Each of the following statements is either true or false. If a statement is true, prove it. If a statement is false, disprove it. These exercises are cumulative, covering all topics addressed in Chapters \(1-9 .\) If \(n \in \mathbb{Z}\) and \(n^{5}-n\) is even, then \(n\) is even.

Each of the following statements is either true or false. If a statement is true, prove it. If a statement is false, disprove it. These exercises are cumulative, covering all topics addressed in Chapters \(1-9 .\) If \(A, B, C\) and \(D\) are sets, then \((A \times B) \cap(C \times D)=(A \cap C) \times(B \cap D)\).

Each of the following statements is either true or false. If a statement is true, prove it. If a statement is false, disprove it. These exercises are cumulative, covering all topics addressed in Chapters \(1-9 .\) If \(A, B\) and \(C\) are sets, then \(A-(B \cup C)=(A-B) \cup(A-C)\).

Each of the following statements is either true or false. If a statement is true, prove it. If a statement is false, disprove it. These exercises are cumulative, covering all topics addressed in Chapters \(1-9 .\) If \(A\) and \(B\) are sets, then \(\mathscr{P}(A)-\mathscr{P}(B) \subseteq \mathscr{P}(A-B)\).

Each of the following statements is either true or false. If a statement is true, prove it. If a statement is false, disprove it. These exercises are cumulative, covering all topics addressed in Chapters \(1-9 .\) If \(A\) and \(B\) are sets and \(A \cap B=\varnothing\), then \(\mathscr{P}(A)-\mathscr{P}(B) \subseteq \mathscr{P}(A-B)\).

Each of the following statements is either true or false. If a statement is true, prove it. If a statement is false, disprove it. These exercises are cumulative, covering all topics addressed in Chapters \(1-9 .\) There exists a set \(X\) for which \(\mathbb{R} \subseteq X\) and \(\varnothing \in X\).

Each of the following statements is either true or false. If a statement is true, prove it. If a statement is false, disprove it. These exercises are cumulative, covering all topics addressed in Chapters \(1-9 .\) If \(A\) and \(B\) are sets, then \(\mathscr{P}(A) \cap \mathscr{P}(B)=\mathscr{P}(A \cap B)\).

Each of the following statements is either true or false. If a statement is true, prove it. If a statement is false, disprove it. These exercises are cumulative, covering all topics addressed in Chapters \(1-9 .\) Every odd integer is the sum of three odd integers.

Each of the following statements is either true or false. If a statement is true, prove it. If a statement is false, disprove it. These exercises are cumulative, covering all topics addressed in Chapters \(1-9 .\) For all sets \(A\) and \(B\), if \(A-B=\varnothing\), then \(B \neq \varnothing\).

Each of the following statements is either true or false. If a statement is true, prove it. If a statement is false, disprove it. These exercises are cumulative, covering all topics addressed in Chapters \(1-9 .\) There exist prime numbers \(p\) and \(q\) for which \(p-q=1000\).

Each of the following statements is either true or false. If a statement is true, prove it. If a statement is false, disprove it. These exercises are cumulative, covering all topics addressed in Chapters \(1-9 .\) The inequality \(2^{x} \geq x+1\) is true for all positive real numbers \(x\).

Each of the following statements is either true or false. If a statement is true, prove it. If a statement is false, disprove it. These exercises are cumulative, covering all topics addressed in Chapters \(1-9 .\) Suppose \(A, B\) and \(C\) are sets. If \(A=B-C,\) then \(B=A \cup C\).

Each of the following statements is either true or false. If a statement is true, prove it. If a statement is false, disprove it. These exercises are cumulative, covering all topics addressed in Chapters \(1-9 .\) The equation \(x^{2}=2^{x}\) has three real solutions.

Each of the following statements is either true or false. If a statement is true, prove it. If a statement is false, disprove it. These exercises are cumulative, covering all topics addressed in Chapters \(1-9 .\) If \(x, y \in \mathbb{R}\) and \(|x+y|=|x-y|\), then \(y=0\).

Each of the following statements is either true or false. If a statement is true, prove it. If a statement is false, disprove it. These exercises are cumulative, covering all topics addressed in Chapters \(1-9 .\) There exist integers \(a\) and \(b\) for which \(42 a+7 b=1\).

Each of the following statements is either true or false. If a statement is true, prove it. If a statement is false, disprove it. These exercises are cumulative, covering all topics addressed in Chapters \(1-9 .\) If \(n, k \in \mathbb{N}\) and \(\left(\begin{array}{l}n \\ k\end{array}\right)\) is a prime number, then \(k=1\) or \(k=n-1\).

Each of the following statements is either true or false. If a statement is true, prove it. If a statement is false, disprove it. These exercises are cumulative, covering all topics addressed in Chapters \(1-9 .\) In Chapter 5, Exercise 25 asked you to prove that if \(2^{n}-1\) is prime, then \(n\) is prime. Is the converse true?