91Ó°ÊÓ

Is the following proposition true or false? For all integers \(a\) and \(b,\) if \(a b\) is even, then \(a\) is even or \(b\) is even. Justify your conclusion by writing a proof if the proposition is true or by providing a counterexample if it is false.

(a) Let \(n \in \mathbb{N}\) and let \(a \in \mathbb{Z}\). Explain why \(n\) divides \(a\) if and only if \(a \equiv 0(\bmod n)\) (b) Let \(a \in \mathbb{Z}\). Explain why if \(a \neq 0(\bmod 3),\) then \(a \equiv 1(\bmod 3)\) or \(a \equiv 2(\bmod 3)\) (c) Is the following proposition true or false? Justify your conclusion. For each \(a \in \mathbb{Z},\) if \(a \neq 0(\bmod 3),\) then \(a^{2} \equiv 1(\bmod 3)\).

Determine if each of the following statements is true or false. If a statement is true, then write a formal proof of that statement, and if it is false, then provide a counterexample that shows it is false. (a) For each integer \(a\), if there exists an integer \(n\) such that \(a\) divides \((8 n+\) 7) and \(a\) divides \((4 n+1),\) then \(a\) divides 5 . (b) For each integer \(a\), if there exists an integer \(n\) such that \(a\) divides \((9 n+\) 5) and \(a\) divides \((6 n+1),\) then \(a\) divides 7 . (c) For each integer \(n,\) if \(n\) is odd, then 8 divides \(\left(n^{4}+4 n^{2}+11\right)\). (d) For each integer \(n,\) if \(n\) is odd, then 8 divides \(\left(n^{4}+n^{2}+2 n\right)\).

Consider the following proposition: For each integer \(a, a \equiv 3(\bmod 7)\) if and only if \(\left(a^{2}+5 a\right) \equiv 3(\bmod 7)\). (a) Write the proposition as the conjunction of two conditional statements. (b) Determine if the two conditional statements in Part (a) are true or false. If a conditional statement is true, write a proof, and if it is false, provide a counterexample. (c) Is the given proposition true or false? Explain.

(Exercise (17), Section 3.2) Let \(a\) and \(b\) be natural numbers such that \(a^{2}=\) \(b^{3} .\) Prove each of the propositions in Parts (6a) through (6d). (The results of Exercise (1) and Theorem 3.10 from Section 3.2 may be helpful.) (a) If \(a\) is even, then 4 divides \(a\). (b) If 4 divides \(a,\) then 4 divides \(b\). (c) If 4 divides \(b,\) then 8 divides \(a\). (d) If \(a\) is even, then 8 divides \(a\). (e) Give an example of natural numbers \(a\) and \(b\) such that \(a\) is even and \(a^{2}=b^{3},\) but \(b\) is not divisible by 8.

Are the following statements true or false? Justify each conclusion. (a) For each positive real number \(x,\) if \(x\) is irrational, then \(x^{2}\) is irrational. (b) For each positive real number \(x\), if \(x\) is irrational, then \(\sqrt{x}\) is irrational. (c) For every pair of real numbers \(x\) and \(y\), if \(x+y\) is irrational, then \(x\) is irrational and \(y\) is irrational. (d) For every pair of real numbers \(x\) and \(y\), if \(x+y\) is irrational, then \(x\) is irrational or \(y\) is irrational.

Are the following propositions true or false? Justify all your conclusions. If a biconditional statement is found to be false, you should clearly determine if one of the conditional statements within it is true. In that case, you should state an appropriate theorem for this conditional statement and prove it. (a) For all integers \(m\) and \(n, m\) and \(n\) are consecutive integers if and only if 4 divides \(\left(m^{2}+n^{2}-1\right)\) (b) For all integers \(m\) and \(n, 4\) divides \(\left(m^{2}-n^{2}\right)\) if and only if \(m\) and \(n\) are both even or \(m\) and \(n\) are both odd.

(a) Give an example that shows that the sum of two irrational numbers can be a rational number. (b) Now explain why the following proof that \((\sqrt{2}+\sqrt{5})\) is an irrational number is not a valid proof: Since \(\sqrt{2}\) and \(\sqrt{5}\) are both irrational numbers, their sum is an irrational number. Therefore, \((\sqrt{2}+\sqrt{5})\) is an irrational number. Note: You may even assume that we have proven that \(\sqrt{5}\) is an irrational number. (We have not proven this.) (c) Is the real number \(\sqrt{2}+\sqrt{5}\) a rational number or an irrational number? Justify your conclusion.

Consider the following proposition: For each integer \(a, a \equiv 2(\bmod 8)\) if and only if \(\left(a^{2}+4 a\right) \equiv 4(\bmod 8)\) (a) Write the proposition as the conjunction of two conditional statements. (b) Determine if the two conditional statements in Part (a) are true or false. If a conditional statement is true, write a proof, and if it is false, provide a counterexample. (c) Is the given proposition true or false? Explain.

For a right triangle, suppose that the hypotenuse has length \(c\) feet and the lengths of the sides are \(a\) feet and \(b\) feet. (a) What is a formula for the area of this right triangle? What is an isosceles triangle? (b) State the Pythagorean Theorem for right triangles. \(\star\) (c) Prove that the right triangle described above is an isosceles triangle if and only if the area of the right triangle is \(\frac{1}{4} c^{2}\).