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Mathematics LibreTexts

6.5: Exercise

  • Page ID
    24832
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    A. Use the method of proof by contradiction to prove the following statements. (In each case, you should also think about how a direct or contrapositive proof would work. You will find in most cases that proof by contradiction is easier.)

    Exercise \(\PageIndex{1}\)

    Suppose \(n \in \mathbb{Z}\). If n is odd, then \(n^2\) is odd.

    Exercise \(\PageIndex{2}\)

    Suppose \(n \in \mathbb{Z}\). If \(n^2\) is odd, then n is odd.

    Exercise \(\PageIndex{3}\)

    Prove that \(\sqrt[3]{2}\) is irrational.

    Exercise \(\PageIndex{4}\)

    Prove that \(\sqrt{6}\) is irrational.

    Exercise \(\PageIndex{5}\)

    Prove that \(\sqrt{3}\) is irrational.

    Exercise \(\PageIndex{6}\)

    If \(a, b \in \mathbb{Z}\), then \(a^2-4b-2 \ne 0\).

    Exercise \(\PageIndex{7}\)

    If \(a, b \in \mathbb{Z}\), then \(a^2-4b-3 \ne 0\).

    Exercise \(\PageIndex{8}\)

    Suppose \(a, b, c \in \mathbb{Z}\). If \(a^2+b^2 = c^2\), then a or b is even.

    Exercise \(\PageIndex{9}\)

    Suppose \(a, b \in \mathbb{R}\). If a is rational and ab is irrational, then b is irrational.

    Exercise \(\PageIndex{10}\)

    There exist no integers a and b for which \(21a + 30b = 1\).

    Exercise \(\PageIndex{11}\)

    There exist no integers a and b for which \(18a + 6b = 1\).

    Exercise \(\PageIndex{12}\)

    For every positive \(x \in \mathbb{Q}\), there is a positive \(y \in \mathbb{Q}\) for which \(y < x\).

    Exercise \(\PageIndex{13}\)

    For every \(x \in [\pi/2, \pi]\), \(sinx-cosx \ge 1\).

    Exercise \(\PageIndex{14}\)

    If A and B are sets, then \(A \cap (B-A)= \emptyset\).

    Exercise \(\PageIndex{15}\)

    If \(b \in \mathbb{Z}\) and \(b|k\) for every \(k \in \mathbb{N}\), then \(b = 0\).

    Exercise \(\PageIndex{16}\)

    If a and b are positive real numbers, then \(a+b \ge 2\sqrt{ab}\).

    Exercise \(\PageIndex{17}\)

    For every \(n \in \mathbb{Z}\), \(4 \nmid (n^2+2)\).

    Exercise \(\PageIndex{18}\)

    Suppose \(a, b \in \mathbb{Z}\). If \(4|(a^2+b^2)\), then a and b are not both odd.

    B. Prove the following statements using any method from Chapters 4, 5 or 6.

    Exercise \(\PageIndex{19}\)

    The product of any five consecutive integers is divisible by 120. (For example, the product of 3,4,5,6 and 7 is 2520, and \(2520 = 120 \cdot 21\).)

    Exercise \(\PageIndex{20}\)

    We say that a point P = (x, y) in \(\mathbb{R}^2\) is rational if both x and y are rational. More precisely, P is rational if \(P =(x,y) \in \mathbb{Q}^2\). An equation \(F(x,y)=0\) is said to have a rational point if there exists \(x_{0}, y_{0} \in \mathbb{Q}\) such that \(F(x_{0}, y_{0}) = 0\). For example, the curve \(x^2+y^2-1 = 0\) has rational point \((x_{0}, y_{0}) = (1, 0)\). Show that the curve \(x^2+y^2-3 = 0\) has no rational points.

    Exercise \(\PageIndex{21}\)

    Exercise 20 (above) involved showing that there are no rational points on the curve \(x^2+y^2-3=0\). Use this fact to show that \(\sqrt{3}\) is irrational.

    Exercise \(\PageIndex{22}\)

    Explain why \(x^2+y^2-3 = 0\) not having any rational solutions (Exercise 20) implies \(x^2+y^2-3k = 0\) has no rational solutions for k an odd, positive integer.

    Exercise \(\PageIndex{23}\)

    Use the above result to prove that \(\sqrt{3^k}\) is irrational for all odd, positive k.

    Exercise \(\PageIndex{24}\)

    The number \(log_{2} 3\) is irrational.