집합과 명제 전체 문제

The following exercises contain definitions or results from more advanced mathematics courses. Even though we may not understand all of the terms involved, it is still possible to recognize the structure of the given statements and write a meaningful negation of that statement.

In abstract algebra, a ring

consists of a nonempty set $R$ and two operations called addition and multiplication. A nonzero element $a$ in a ring $R$ is called a zero divisor

provided that there exists a nonzero element $b$ in $R$ such that $ab = 0$ or $ba = 0$. Carefully explain what it means to say that a nonzero element $a$ in a ring $R$ is not a zero divisor.

Give at least four different examples of integers $a$, $b$, and $c$ with $a \ne 0$ such that $a$ divides $b$ and $a$ divides $c$.

Complete the following.

If $x$ and $y$ are integers and $xy=1$, explain why $x=1$ or $x=-1$.

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

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 $\left(a, b \right)$.

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 $ab \ne 0$.”

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 $a \ne b$.

Consider the following statement:

For each positive real number

r

, if

r^2 = 18

, then

r

is irrational.

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.

Complete the following.

Give an example that shows that the sum of two irrational numbers can be a rational number.

Complete the following.

Now explain why the following proof that $\left( \sqrt{2} + \sqrt{5} \right)$ 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, $\left( \sqrt{2} + \sqrt{5} \right)$ 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.)

Three natural numbers $a$, $b$, and $c$ with $a \lt b \lt c$ are called a Pythagorean triple

provided that $a^2 + b^2 = c^2$. For example, the numbers 3, 4, and 5 form a Pythagorean triple, and the numbers 5, 12, and 13 form a Pythagorean triple.

Determine two other Pythagorean triples. That is, find integers $a, b$, and $c$ such that $a^2 + b^2 = c^2$.

Consider the following proposition: There are no integers $a$ and $b$ such that $b^2 = 4a + 2$.

Rewrite this statement in an equivalent form using a universal quantifier by completing the following:

For all integers

a

and

b

,

.

Let the universal set be $U = \left\{ {1,2,3,4,5,6} \right\}$, and let

$$A = \left\{ {1,2,4} \right\}\!, B = \left\{ {1,2,3,5} \right\}\!, C = \left\{ {\left. {x \in U} \right|x^2 \leq 2} \right\}\!$$

.

In each of the following, fill in the blank with one or more of the symbols $\subset, \\ \subseteq, =, \ne, \in ,\text{ or } \notin$ so that the resulting statement is true. For each blank, include all symbols that result in a true statement. If none of these symbols makes a true statement, write nothing in the blank.

$A$_____$B$

Let the universal set be $U = \left\{ {1,2,3,4,5,6} \right\}$, and let

$$A = \left\{ {1,2,4} \right\}\!, B = \left\{ {1,2,3,5} \right\}\!, C = \left\{ {\left. {x \in U} \right|x^2 \leq 2} \right\}\!$$

.

In each of the following, fill in the blank with one or more of the symbols $\subset, \\ \subseteq, =, \ne, \in ,\text{ or } \notin$ so that the resulting statement is true. For each blank, include all symbols that result in a true statement. If none of these symbols makes a true statement, write nothing in the blank.

5_____$B$

Let the universal set be $U = \left\{ {1,2,3,4,5,6} \right\}$, and let

$$A = \left\{ {1,2,4} \right\}\!, B = \left\{ {1,2,3,5} \right\}\!, C = \left\{ {\left. {x \in U} \right|x^2 \leq 2} \right\}\!$$

.

In each of the following, fill in the blank with one or more of the symbols $\subset, \\ \subseteq, =, \ne, \in ,\text{ or } \notin$ so that the resulting statement is true. For each blank, include all symbols that result in a true statement. If none of these symbols makes a true statement, write nothing in the blank.

$A$_____$C$

Let the universal set be $U = \left\{ {1,2,3,4,5,6} \right\}$, and let

$$A = \left\{ {1,2,4} \right\}\!, B = \left\{ {1,2,3,5} \right\}\!, C = \left\{ {\left. {x \in U} \right|x^2 \leq 2} \right\}\!$$

.

In each of the following, fill in the blank with one or more of the symbols $\subset, \\ \subseteq, =, \ne, \in ,\text{ or } \notin$ so that the resulting statement is true. For each blank, include all symbols that result in a true statement. If none of these symbols makes a true statement, write nothing in the blank.

$\left\{ {1, 2} \right\}$_____$A$