91Ó°ÊÓ

In Preview Activity \(2,\) we proved that if \(n\) is an integer, then \(n^{2}+n\) is an even integer. We define two integers to be consecutive integers if one of the integers is one more than the other integer. This means that we can represent consecutive integers as \(m\) and \(m+1,\) where \(m\) is some integer.

(Exercise (15), Section 3.1) Let \(r\) be a positive real number. The equation for a circle of radius \(r\) whose center is the origin is \(x^{2}+y^{2}=r^{2}\). (a) Use implicit differentiation to determine \(\frac{d y}{d x}\). (b) (Exercise (17), Section 3.2) Let \((a, b)\) be a point on the circle with \(a \neq 0\) and \(b \neq 0\). Determine the slope of the line tangent to the circle at the point \((a, b)\). (c) Prove that the radius of the circle to the point \((a, b)\) is perpendicular to the line tangent to the circle at the point \((a, b)\). Hint: Two lines (neither of which is horizontal) are perpendicular if and only if the products of their slopes is equal to -1

In Section \(3.1,\) we defined congruence modulo \(n\) where \(n\) is a natural number. If \(a\) and \(b\) are integers, we will use the notation \(a \neq b(\bmod n)\) to mean that \(a\) is not congruent to \(b\) modulo \(n\). * (a) Write the contrapositive of the following conditional statement: For all integers \(a\) and \(b,\) if \(a \neq 0(\bmod 6)\) and \(b \neq 0(\bmod 6),\) then \(a b \not \equiv 0(\bmod 6)\). (b) Is this statement true or false? Explain.

Are the following statements true or false? Justify your conclusions. (a) For each integer \(a\), if 3 does not divide \(a\), then 3 divides \(2 a^{2}+1\). (b) For each integer \(a\), if 3 divides \(2 a^{2}+1,\) then 3 does not divide \(a\). (c) For each integer \(a, 3\) does not divide \(a\) if and only if 3 divides \(2 a^{2}+1\).

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 all integers \(a, b,\) and \(c\) with \(a \neq 0,\) if \(a \mid b,\) then \(a \mid(b c)\). (b) For all integers \(a\) and \(b\) with \(a \neq 0,\) if \(6 \mid(a b),\) then \(6 \mid a\) or \(6 \mid b\). (c) For all integers \(a, b,\) and \(c\) with \(a \neq 0,\) if \(a\) divides \((b-1)\) and \(a\) divides \((c-1),\) then \(a\) divides \((b c-1)\) (d) For each integer \(n,\) if 7 divides \(\left(n^{2}-4\right),\) then 7 divides \((n-2)\). (e) For every integer \(n, 4 n^{2}+7 n+6\) is an odd integer. ? (f) For every odd integer \(n, 4 n^{2}+7 n+6\) is an odd integer. (g) For all integers \(a, b,\) and \(d\) with \(d \neq 0,\) if \(d\) divides both \(a-b\) and \(a+b,\) then \(d\) divides \(a\) (h) For all integers \(a, b,\) and \(c\) with \(a \neq 0,\) if \(a \mid(b c),\) then \(a \mid b\) or \(a \mid c .\)

Prove that the cube root of 2 is an irrational number. That is, prove that if \(r\) is a real number such that \(r^{3}=2,\) then \(r\) is an irrational number.

Prove that for each real number \(x\) and each irrational number \(q,(x+q)\) is irrational or \((x-q)\) is irrational.

Are the following statements true or false? Justify your conclusions. (a) For each \(a \in \mathbb{Z},\) if \(a \equiv 2(\bmod 5),\) then \(a^{2} \equiv 4(\bmod 5)\). (b) For each \(a \in \mathbb{Z},\) if \(a^{2} \equiv 4(\bmod 5),\) then \(a \equiv 2(\bmod 5)\) (c) For each \(a \in \mathbb{Z}, a \equiv 2(\bmod 5)\) if and only if \(a^{2} \equiv 4(\bmod 5)\).

Consider the following statement: For each positive real number \(r,\) if \(r^{2}=18,\) then \(r\) is irrational. (a) If you were setting up a proof by contradiction for this statement, what would you assume? Carefully write down all conditions that you would assume. (b) Complete a proof by contradiction for this statement.

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.