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Magazine Problem 24
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 \(x\).
Problem 26
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 \(B=A \cup C\).
Problem 27
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.
Problem 29
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 \(y=0\).
Problem 30
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 \(42 a+7 b=1\).
Problem 32
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 \(k=n-1\).
Problem 35
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?
