명제의 증명과 절대부등식

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.

For each integer $a$, if there exists an integer $n$ such that $a$ divides $(9n + 5)$ and $a$ divides $(6n + 1)$, then $a$ divides 7.

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.

For each integer $n$, if $n$ is odd, then 8 divides $\left( n^4 + 4n^2 + 11 \right)$.

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.

For each integer $n$, if $n$ is odd, then 8 divides $\left( n^4 + n^2 + 2n \right)$.

Complete the following.

Prove that if $a$, $b$, and $c$ are real numbers for which $a>0$ and $c\lt 0$, then one solution of the quadratic equation $ax^2+bx+c=0$ is a positive real number.

Complete the following.

Prove that if $a$, $b$, and $c$ are real numbers, if $a \ne 0$, $b > 0$ and $\dfrac{b}{2} \lt \sqrt {ac}$, then the quadratic equation $ax^2 + bx + c = 0$ has no real number solution.

Let $h$ and $k$ be real numbers and let $r$ be a positive number. The equation for a circle whose center is at the point $\left( h, k \right)$ and whose radius is $r$ is

$$\left( x - h \right)^2 + \left( y - k \right)^2 = r^2$$

. We also know that if $a$ and $b$ are real numbers, then

• The point $\left( a, b \right)$ is inside the circle if $\left( a - h \right)^2 + \left( b - k \right)^2 \lt r^2$. - The point $\left( a, b \right)$ is on the circle if $\left( a - h \right)^2 + \left( b - k \right)^2 = r^2$. - The point $\left( a, b \right)$ is outside the circle if $\left( a - h \right)^2 + \left( b - k \right)^2 > r^2$.

Prove that all points on or inside the circle whose equation is $\left( x - 1 \right)^2 + \left( y - 2 \right)^2 = 4$ are inside the circle whose equation is $x^2 + y^2 = 26$.

Let $r$ be a positive real number. The equation for a circle of radius $r$ whose center is the origin is $x^2 + y^2 = r^2$.

Use implicit differentiation to determine $\dfrac{dy}{dx}$.

Let $r$ be a positive real number. The equation for a circle of radius $r$ whose center is the origin is $x^2 + y^2 = r^2$.

Let $\left(a, b \right)$ be a point on the circle with $a \ne 0$ and $b \ne 0$. Determine the slope of the line tangent to the circle at the point $\left(a, b \right)$.

Let $r$ be a positive real number. The equation for a circle of radius $r$ whose center is the origin is $x^2 + y^2 = r^2$.

Prove that the radius of the circle to the point $\left(a, b \right)$ is perpendicular to the line tangent to the circle at the point $\left(a, b \right)$.

Hint: Two lines (neither of which is horizontal) are perpendicular if and only if the products of their slopes is equal to

-1

.

Determine if each of the following statements is true or false. Provide a counterexample for statements that are false and provide a complete proof for those that are true.

For all real numbers $x$ and $y$, $\sqrt{xy} \leq \dfrac{x + y}{2}$.

Determine if each of the following statements is true or false. Provide a counterexample for statements that are false and provide a complete proof for those that are true.

For all real numbers $x$ and $y$, $xy \leq \left( \dfrac{x + y}{2} \right)^2$.

Determine if each of the following statements is true or false. Provide a counterexample for statements that are false and provide a complete proof for those that are true.

For all nonnegative real numbers $x$ and $y$, $\sqrt{xy} \leq \dfrac{x + y}{2}$.

This type of exercise will appear frequently in the book. In each case, there is a proposed proof of a proposition. However, the proposition may be true or may be false.

• If a proposition is false, the proposed proof is, of course, incorrect. In this situation, you are to find the error in the proof and then provide a counterexample showing that the proposition is false. - If a proposition is true, the proposed proof may still be incorrect. In this case, you are to determine why the proof is incorrect and then write a correct proof using the writing guidelines that have been presented in this book. - If a proposition is true and the proof is correct, you are to decide if the proof is well written or not. If it is well written, then you simply must indicate that this is an excellent proof and needs no revision. On the other hand, if the proof is not well written, then you must then revise the proof by writing it according to the guidelines presented in this text.

• Proposition If $m$ is an even integer, then $\left(5m + 4\right)$ is an even integer. - Proof We see that $5m + 4 = 10n + 4 = 2 \left(5n + 2 \right)$. Therefore, $\left(5m + 4 \right)$ is an even integer.

This type of exercise will appear frequently in the book. In each case, there is a proposed proof of a proposition. However, the proposition may be true or may be false.

• If a proposition is false, the proposed proof is, of course, incorrect. In this situation, you are to find the error in the proof and then provide a counterexample showing that the proposition is false. - If a proposition is true, the proposed proof may still be incorrect. In this case, you are to determine why the proof is incorrect and then write a correct proof using the writing guidelines that have been presented in this book. - If a proposition is true and the proof is correct, you are to decide if the proof is well written or not. If it is well written, then you simply must indicate that this is an excellent proof and needs no revision. On the other hand, if the proof is not well written, then you must then revise the proof by writing it according to the guidelines presented in this text.

• Proposition For all real numbers $x$ and $y$, if $x \ne y$, $x > 0$, and $y >0$, then $\dfrac{x}{y} + \dfrac{y}{x} > 2$. - Proof Since $x$ and $y$ are positive real numbers, $xy$ is positive and we can multiply both sides of the inequality by $xy$ to obtain $$\begin{aligned}\left( \frac{x}{y} + \frac{y}{x} \right) \cdot xy & > 2 \cdot xy \\ x^2 + y^2 & > 2xy\end{aligned}$$ . By combining all terms on the left side of the inequality, we see that $x^2 - 2xy + y^2 > 0$ and then by factoring the left side, we obtain $\left( x - y \right)^2 > 0$. Since $x \ne y$, $\left(x - y \right) \ne 0$ and so $\left( x - y \right)^2 > 0$. This proves that if $x \ne y$, $x > 0$, and $y >0$, then $\dfrac{x}{y} + \dfrac{y}{x} > 2$.

This type of exercise will appear frequently in the book. In each case, there is a proposed proof of a proposition. However, the proposition may be true or may be false.

• If a proposition is false, the proposed proof is, of course, incorrect. In this situation, you are to find the error in the proof and then provide a counterexample showing that the proposition is false. - If a proposition is true, the proposed proof may still be incorrect. In this case, you are to determine why the proof is incorrect and then write a correct proof using the writing guidelines that have been presented in this book. - If a proposition is true and the proof is correct, you are to decide if the proof is well written or not. If it is well written, then you simply must indicate that this is an excellent proof and needs no revision. On the other hand, if the proof is not well written, then you must then revise the proof by writing it according to the guidelines presented in this text.

• Proposition For all integers $a$, $b$, and $c$, if $a \mid \left( bc \right)$, then $a \mid b$ or $a \mid c$. - Proof We assume that $a$, $b$, and $c$ are integers and that $a$ divides $bc$. So, there exists an integer $k$ such that $bc = ka$. We now factor $k$ as $k = mn$, where $m$ and $n$ are integers. We then see that $$bc = mna$$ . This means that $b = ma$ or $c = na$ and hence, $a \mid b$ or $a \mid c$.

This type of exercise will appear frequently in the book. In each case, there is a proposed proof of a proposition. However, the proposition may be true or may be false.

• If a proposition is false, the proposed proof is, of course, incorrect. In this situation, you are to find the error in the proof and then provide a counterexample showing that the proposition is false. - If a proposition is true, the proposed proof may still be incorrect. In this case, you are to determine why the proof is incorrect and then write a correct proof using the writing guidelines that have been presented in this book. - If a proposition is true and the proof is correct, you are to decide if the proof is well written or not. If it is well written, then you simply must indicate that this is an excellent proof and needs no revision. On the other hand, if the proof is not well written, then you must then revise the proof by writing it according to the guidelines presented in this text.

• Proposition For all positive integers $a$, $b$, and $c$, $\left( a^b \right)^c = a^{\left( b^c \right)}$. - Proof This proposition is false as is shown by the following counterexample: If we let $a = 2$, $b = 3$, and $c = 2$, then $$\begin{aligned}\left( a^b \right)^c & = a^{\left( b^c \right)} \\ \left( 2^3 \right)^2 & = 2^{\left( 3^2 \right)} \\ 8^2 & = 2^9 \\ 64 & \ne 512\end{aligned}$$ .

In English, write the contrapositive of, “For all real numbers $a$ and $b$, if $a \ne 0$ and $b \ne 0$, then $ab \ne 0$.”

Let $n$ be an integer. Prove each of the following:

If $n$ is even, then $n^3$ is even.

Hint: Let $n$ be an even integer. Since $n$ is even, there exists an integer $k$ such that $n = 2k$. Now use this to prove that $n^3$ must be even.

Let $n$ be an integer. Prove each of the following:

If $n^3$ is even, then $n$ is even.

Hint: Prove the contrapositive.

Complete the following.

Write the contrapositive of the following statement: For all positive real numbers $a$ and $b$, if $\sqrt {ab} \ne \dfrac{{a + b}}{2}$, then $a \ne b$.