1183 lines
29 KiB
Markdown
1183 lines
29 KiB
Markdown
**Exercise Set 1.1**
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Page 28
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In each of 1-6, fill in the blanks using a variable or variables to rewrite the
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given statement.
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1. Is there a real number whose square is $-1$?
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a. Is there a real number $x$ such that ______?
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b. Does there exist ______ such that $x^2 = -1$?
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**Solution**
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a. Is there a real number $x$ such that $x^2 = -1$?
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b. Does there exist a real number $x$ such that $x^2 = -1$?
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2. Is there an integer that has a remainder of $2$ when it is divided by $5$ and
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a remainder of $3$ when it is divided by $6$?
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a. Is there an integer $n$ such that $n$ has ______?
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b. Does there exist ______ such that if $n$ is divided by $5$ the remainder is
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$2$ and if ______?
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_Note: There are integers with this property. Can you think of one?_
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**Solution**
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a. Is there an integer $n$ such that $n$ has a remainder of $2$ when $n$ is
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divided by $5$ and a remainder of $3$ when $n$ is divided by $6$?
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b. Does there exist a number $n$ such that if $n$ is divided by $5$ the
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remainder is $2$ and if $n$ is divided by $6$ the remainder is $3$?
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_Note: There are integers with this property. Can you think of one?_
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$$ 27 \mod 5 = 2 $$
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$$ 27 \mod 6 = 3 $$
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3. Given any two distinct real numbers, there is a real number in between them.
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a. Given any two distinct real numbers $a$ and $b$, there is a real number $c$
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such that $c$ is ______.
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b. For any two ______, ______ such that $c$ is between $a$ and $b$.
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**Solution**
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a. Given any two distinct real numbers $a$ and $b$, there is a real number $c$
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such that $c$ is $a \leq c \leq b$.
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b. For any two distinct real numbers $a$ and $b$, there exists a real number $c$
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such that $c$ is between $a$ and $b$.
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4. Given any real number, there is a real number that is greater.
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a. Given any real number $r$, there is ______ $s$ such that $s$ is ______
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b. For any ______, ______ such that $s > r$.
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**Solution**
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a. Given any real number $r$, there is a real number $s$ such that $s$ is
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greater than $r$.
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b. For any real number $r$, there exists a real number $s$ such that $s > r$.
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5. The reciprocal of any positive real number is positive.
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a. Given any positive real number $r$, the reciprocal of ______.
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b. For any real number $$, if $r$ is ______, then ______.
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c. If a real number $r$ ______, then ______.
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**Solution**
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a. Given any positive real number $r$, the reciprocal of $r$ is positive.
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b. For any real number $r$, if $r$ is positive, then the reciprocal of $r$ is
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positive.
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c. If a real number $r$ is positive, then the reciprocal of $r$ is positive.
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6. The cube root of any negative real number is negative.
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a. Given any negative real number $s$, the cube root of ______.
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b. For any real number $s$, if $s$ is ______, then ______.
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c. If a real number $s$ ______, then ______.
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**Solution**
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a. Given any negative real number $s$, the cube root of $s$ is negative.
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b. For any real number $s$, if $s$ is negative, then the cube root of $s$ is
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negative.
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c. If a real number $s$ is negative, then the cube root of $s$ is negative.
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7. Rewrite the following statements less formally, without using variables.
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Determine, as best as you can, whether the statements are true or false.
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a. There are real numbers $u$ and $v$ with the property that $u + v < u - v$.
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b. There is a real number $x$ such that $x^2 < x$.
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c. For every positive integer $n$, $n^2 \geq n$.
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d. For all real numbers $a$ and $b$, $|a + b| \leq |a| + |b|$.
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**Solution**
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a. There are real numbers $u$ and $v$ with the property that $u + v < u - v$.
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There are two distinct real numbers where the sum of those two numbers is less
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than the difference of those two numbers.
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This is true if you consider our domain is all real numbers which include
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negatives. For example:
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$$ 1 + (-1) = 0 $$
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$$ 1 - (-1) = 2 $$
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$$ 0 < 2 $$
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b. There is a real number $x$ such that $x^2 < x$.
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There is a real number which is greater than it's square.
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This is true for any fraction/decimal. Consider:
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$$ \left(\frac{1}{4}\right)^2 = \frac{1}{16} $$
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$$ \frac{1}{16} < \frac{1}{4} $$
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c. For every positive integer $n$, $n^2 \geq n$.
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For all positive integers, an integer's square is always greater than or equal
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to the integer.
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This is true. Starting at $1$ we get $1^2 \geq 1$, which is true, $2^2 \geq 2$
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is true, and so on. We're essentially multiplying each side of the inequality by
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some positive integer, which we know from algebra does not change the direction
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of the inequality, so this statement holds true.
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d. For all real numbers $a$ and $b$, $|a + b| \leq |a| + |b|$.
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For any two distinct real numbers, the absolute value of their sum is less than
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or equal to the sum of the absolute values of each number.
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This is true, if both $a$ and $b$ are positive numbers or both $a$ and $b$ are
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negative integers, then the two statements are equal. If either $a$ or $b$ is
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negative and the other is positive, then the left statement will always be less
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than the right hand statement.
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---
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In each of 8-13, fill in the blanks to rewrite the given statement.
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8. For every object $J$, if $J$ is a square then $J$ has four sides.
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a. All squares ______.
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b. Every square ______.
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c. If an object is a square, then it ______.
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d. If $J$ ______, then $J$ ______.
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e. For every square $J$, ______.
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**Solution**
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a. All squares have four sides.
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b. Every square has four sides.
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c. If an object is a square, then it has four sides.
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d. If $J$ is a square, then $J$ has four sides.
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e. For every square $J$, $J$ has four sides.
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9. For every equation $E$, if $E$ is quadratic then $E$ has at most two real
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solutions.
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a. All quadratic equations ______.
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b. Every quadratic equation ______.
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c. If an equation is quadratic, then it ______.
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d. If $E$ ______, then $E$ ______.
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e. For every quadratic equation $E$, ______.
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**Solution**
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a. All quadratic equations have at most two real solutions.
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b. Every quadratic equation has at most two real solutions.
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c. If an equation is quadratic, then it has at most two real solutions.
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d. If $E$ is a quadratic equation, then $E$ has at most two real solutions.
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e. For every quadratic equation $E$, $E$ has at most two real solutions.
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10. Every nonzero real number has a reciprocal.
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a. All nonzero real numbers ______.
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b. For every nonzero real number $r$, there is ______ for $r$.
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c. For every nonzero real number $r$, there is a real number $s$ such that
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______.
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**Solution**
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a. All nonzero real numbers have reciprocals.
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b. For every nonzero real number $r$, there is a reciprocal for $r$.
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c. For every nonzero real number $r$, there is a real number $s$ such that $s$
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is a reciprocal of $r$.
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11. Every positive number has a positive square root.
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a. All positive numbers ______.
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b. For every positive number $e$, there is ______ for $e$.
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c. For every positive number $e$, there is a positive number $r$ such that
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______.
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**Solution**
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a. All positive numbers have positive square roots.
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b. For every positive number $e$, there is a positive square root for $e$.
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c. For every positive number $e$, there is a positive number $r$ such that $r$
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is a positive square root for $e$.
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12. There is a real number whose product with every number leaves the number
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unchanged.
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a. Some ______ has the property that its ______.
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b. There is a real number $r$ such that the product of $r$ ______.
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c. There is a real number $r$ with the property that for every real number $s$,
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______.
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**Solution**
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a. Some real number has the property that its product with every number leaves
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the number unchanged.
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b. There is a real number $r$ such that the product of $r$ with every number
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leaves $r$ unchanged.
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c. There is a real number $r$ with the property that for every real number $s$,
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such that $rs = s$.
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13. There is a real number whose product with every real number equals zero.
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a. Some _____ has the property that its ______.
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b. There is a real number $a$ such that the product of $a$ ______.
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c. There is a real number $a$ with the property that for every real number $b$,
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______.
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**Solution**
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a. Some real number has the property that its product with every real number
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equals zero.
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b. There is a real number $a$ such that the product of $a$ with every real
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number equals zero.
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c. There is a real number $a$ with the property that for every real number $b$,
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$ab = 0$.
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---
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**Exercise Set 1.2**
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Page 37
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1. Which of the following sets are equal?
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$$ A = \{a, b, c, d\} $$
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$$ B = \{d, e, a, c\} $$
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$$ C = \{d, b, a, c\} $$
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$$ D = \{a, a, d, e, c, e\} $$
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**Solution**
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$$ A = C $$
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$$ B = D $$
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2. Write in words how to read each of the following out loud.
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a. $\{x \in \mathbb{R}^+ | 0 < x < 1\}$
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b. $\{x \in \mathbb{R} | x \leq 0 \text{ or } x \geq 1\}$
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c. $\{n \in \mathbb{Z} | n \text{ is a factor of } 6\}$
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d. $\{n \in \mathbb{Z}^+ | n \text{ is a factor of } 6\}$
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**Solution**
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a. $\{x \in \mathbb{R}^+ | 0 < x < 1\}$
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The set of all positive real numbers $x$ such that $x$ is greater than $0$ and
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less than $1$.
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b. $\{x \in \mathbb{R} | x \leq 0 \text{ or } x \geq 1\}$
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The set of all real numbers $x$ such that $x$ is less than or equal to $0$ or
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$x$ is greater than or equal to $1$.
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c. $\{n \in \mathbb{Z} | n \text{ is a factor of } 6\}$
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The set of all integers $n$ such that $n$ is a factor of $6$.
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d. $\{n \in \mathbb{Z}^+ | n \text{ is a factor of } 6\}$
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The set of all positive integers $n$ such that $n$ is a factor of $6$.
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3.
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a. Is $4 = \{4\}$?
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b. How many elements are in the set $\{3, 4, 3, 5\}$?
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c. How many elements are in the set $\{1, \{1\}, \{1, \{1\}\}\}$ ?
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**Solution**
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a. Is $4 = \{4\}$?
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No, the symbol $4$, which represents the number four, does not equal the set
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that contains an element that is the number $4$.
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b. How many elements are in the set $\{3, 4, 3, 5\}$?
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There are 3 elements in the set $\{3, 4, 3, 5\}$. Repeated elements are not
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counted as more than 1 element in a set.
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c. How many elements are in the set $\{1, \{1\}, \{1, \{1\}\}\}$ ?
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There are three elements in the set, namely the symbol $1$, the set $\{1\}$, and
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the set $\{1, \{1\}\}$.
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4.
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a. Is $2 \in \{2}$ ?
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b. How many elements are in the set $\{2, 2, 2, 2\}$ ?
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c. How many elements are in the set $\{0, \{0\}\}$ ?
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d. Is $\{0\} \in \{\{0\}, \{1\}\}$ ?
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e. Is $0 \in \{\{0\}, \{1\}\}$ ?
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**Solution**
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a. Is $2 \in \{2}$ ?
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No, the symbol $2$ which represents the number two, is not equal to the set
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$\{2\}$, which is a set that contains the element $2$.
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b. How many elements are in the set $\{2, 2, 2, 2\}$ ?
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There is one element in the set $\{2, 2, 2, 2\}$, namely the element $2$.
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c. How many elements are in the set $\{0, \{0\}\}$ ?
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There are two elements in the set, namely the symbol $0$, and the set $\{0\}$.
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d. Is $\{0\} \in \{\{0\}, \{1\}\}$ ?
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Yes, the set of $\{0\}$ is in the set $\{\{0\}, \{1\}\}$, as the set contains
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both the sets $\{0\}$ and $\{1\}$.
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e. Is $0 \in \{\{0\}, \{1\}\}$ ?
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No, the symbol $0$, representing the number zero, is not in the set, which holds
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two sets with the symbols in them.
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5. Which of the following sets are equal?
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$$
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A = \{0, 1, 2\} \\
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B = \{x \in \mathbb{R} | -1 \leq x < 3\} \\
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C = \{x \in \mathbb{R} | -1 < x < 3\} \\
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D = \{x \in \mathbb{Z} | -1 < x < 3\} \\
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E = \{x \in \mathbb{Z}^+ | -1 < x < 3\}
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$$
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**Solution**
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None of these sets are equal. $A = E$ might have worked had $A$ not included
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$0$, but $E$ essentially evaluates to $E = \{1, 2\}$, and does not include $0$.
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6. For each integer $n$, let $T_n = \{n, n^2\}$. How many elements are in each
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of $T_2, T_{-3}, T_1$, and $T_0$? Justify your answers.
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**Solution**
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$$ T_2 = \{2, 2^2\} = \{2, 4\} \quad \text{ two elements } $$
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$$ T_{-3} = \{-3, (-3)^2\} = \{-3, 9\} \quad \text{ two elements }$$
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$$ T_1 = \{1, 1^2\} = \{1, 1\} = \{1\} \quad \text{ one element } $$
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$$ T_0 = \{0, 0^2\} = \{0, 0\} = \{0\} \quad \text{ one element } $$
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7. Use the set-roster notation to indicate the elements in each of the following
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sets.
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a. $S = \{n \in \mathbb{Z} | n = (-1)^k \text{, for some integer } k\}$
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b. $T = \{m \in \mathbb{Z}| m = 1 + (-1)^i \text{, for some integer } i\}$
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c. $U = \{r \in \mathbb{Z} | 2 \leq r \leq -2\}$
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d. $V = \{s \in \mathbb{Z} | s > 2 \text{ or }x < 3\}$
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e. $W = \{t \in \mathbb{Z} | 1 < t < -3\}$
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f. $X = \{u \in \mathbb{Z} | u \leq 4 \text{ or } u \geq 1\}$
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**Solution**
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a. $S = \{n \in \mathbb{Z} | n = (-1)^k \text{, for some integer } k\}$
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$$ \{-1, 1\} $$
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b. $T = \{m \in \mathbb{Z}| m = 1 + (-1)^i \text{, for some integer } i\}$
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$$ \{0, 2\} $$
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c. $U = \{r \in \mathbb{Z} | 2 \leq r \leq -2\}$
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$$ \emptyset $$
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d. $V = \{s \in \mathbb{Z} | s > 2 \text{ or }x < 3\}$
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$$ \mathbb{Z} $$
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e. $W = \{t \in \mathbb{Z} | 1 < t < -3\}$
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$$ \emptyset $$
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f. $X = \{u \in \mathbb{Z} | u \leq 4 \text{ or } u \geq 1\}$
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$$ \mathbb{Z}^+ $$
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8. Let $A = \{c, d, f, g\}$, $B = \{f, j\}$, and $C = \{d, g\}$. Answer each of
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the following questions. Give reasons for your answers.
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a. Is $B \subseteq A$?
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b. Is $C \subseteq A$?
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c. Is $C \subseteq C$?
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d. Is $C$ a proper subset of $A$?
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**Solution**
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a. Is $B \subseteq A$?
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No, because every element of $B$ must be an element of $A$ by definition of a
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subset, but $j \in B$, but $j \notin A$.
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b. Is $C \subseteq A$?
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Yes, every element of $C$ is an element of $A$.
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c. Is $C \subseteq C$?
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Yes, every element of $C$ is an element of $C$. By implication, every set is a
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subset of itself.
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d. Is $C$ a proper subset of $A$?
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Yes, $C \subset A$, but $C \neq A$. Every element of $C$ is an element of $A$,
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but $C$ does not equal $A$, which is the definition of a proper subset.
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9.
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a. Is $3 \in \{1, 2, 3\}$?
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b. Is $1 \subseteq \{1}$?
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c. Is $\{2\} \in \{1, 2\}$?
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d. Is $\{3\} \in \{1, \{2\}, \{3\}\}$?
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e. Is $1 \in \{1\}$?
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f. Is $\{2\} \subseteq \{1, \{2\}, \{3\}\}$?
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g. Is $\{1\} \subseteq \{1, 2\}$?
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h. Is $1 \in \{\{1\}, 2\}$?
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i. Is $\{1\} \subseteq \{1, \{2\}\}$?
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j. Is $\{1\} \subseteq \{1\}$?
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**Solution**
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a. Is $3 \in \{1, 2, 3\}$?
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Yes, the symbol $3$, representing the number three, is in the set $\{1, 2, 3\}$.
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b. Is $1 \subseteq \{1}$?
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No, the number $1$ is not a set, and therefore cannot be a subset of $\{1\}$.
|
|
|
|
c. Is $\{2\} \in \{1, 2\}$?
|
|
|
|
No, the subset $\{2\}$ is not in the set $\{1, 2\}$, the number $2$ is in the
|
|
subset, but not the set $\{2\}$.
|
|
|
|
d. Is $\{3\} \in \{1, \{2\}, \{3\}\}$?
|
|
|
|
Yes, the set $\{3\}$ is an element of $\{1, \{2\}, \{3\}\}$.
|
|
|
|
e. Is $1 \in \{1\}$?
|
|
|
|
Yes, the number $1$ is in the set $\{1\}$.
|
|
|
|
f. Is $\{2\} \subseteq \{1, \{2\}, \{3\}\}$?
|
|
|
|
No, the set $\{2\}$ holds the element $2$, and $2$ is not an element in
|
|
$\{1, \{2\}, \{3\}\}$.
|
|
|
|
g. Is $\{1\} \subseteq \{1, 2\}$?
|
|
|
|
Yes, the set $\{\1}$ holds the element $1$, and $1$ is an element of $\{1, 2\}$.
|
|
|
|
h. Is $1 \in \{\{1\}, 2\}$?
|
|
|
|
No, the element $1$ is not in $\{\{1\}, 2\}$.
|
|
|
|
i. Is $\{1\} \subseteq \{1, \{2\}\}$?
|
|
|
|
Yes, the set $\{1\}$ holds the element $1$, which is an element of
|
|
$\{1, \{2\}\}$.
|
|
|
|
j. Is $\{1\} \subseteq \{1\}$?
|
|
|
|
Yes $\{1\}$ holds the element $1$, which is an element of $\{1\}$. They are
|
|
equal and it is implied that any set is a subset of itself.
|
|
|
|
10.
|
|
|
|
a. Is $((-2)^2, -2^2) = (-2^2, (-2)^2)$?
|
|
|
|
b. Is $(5, -5) = (-5, 5)$?
|
|
|
|
c. Is $(8 - 9, \sqrt[3]{-1}) = (-1, -1)$?
|
|
|
|
d. Is $\left(\dfrac{-2}{-4}, (-2)^3\right) = \left(\dfrac{3}{6}, -8\right)$
|
|
|
|
**Solution**
|
|
|
|
a. Is $((-2)^2, -2^2) = (-2^2, (-2)^2)$?
|
|
|
|
$$ ((-2)^2, -2^2) = (4, -4) \neq (-4, 4) = (-2^2, (-2)^2) $$
|
|
|
|
So no, they are not equal. For ordered pair tuples to be equal, the order
|
|
matters and so each entry into the tuple must match the other for them to be
|
|
equal.
|
|
|
|
b. Is $(5, -5) = (-5, 5)$?
|
|
|
|
No, ordered pair tuples require that the entries be equal to each other _in
|
|
order_.
|
|
|
|
c. Is $(8 - 9, \sqrt[3]{-1}) = (-1, -1)$?
|
|
|
|
$$ (8 - 9, \sqrt[3]{-1}) = (-1, -1) = (-1, -1) $$
|
|
|
|
So yes, these two ordered pair tuples are equal.
|
|
|
|
d. Is $\left(\dfrac{-2}{-4}, (-2)^3\right) = \left(\dfrac{3}{6}, -8\right)$
|
|
|
|
$$ \left(\dfrac{-2}{-4}, (-2)^3\right) = \left(\frac{1}{2}, -8\right) = \left(\frac{1}{2}, -8\right) = \left(\dfrac{3}{6}, -8\right) $$
|
|
|
|
So yes, these two ordered pair tuples are equal.
|
|
|
|
11. Let $A = \{w, x, y, z\}$ and $B = \{a, b\}$. Use set-roster notation to
|
|
write each of the following sets, and indicate the number of elements that
|
|
are in each set.
|
|
|
|
a. $A \times B$
|
|
|
|
b. $B \times A$
|
|
|
|
c. $A \times A$
|
|
|
|
d. $B \times B$
|
|
|
|
**Solution**
|
|
|
|
a. $A \times B$
|
|
|
|
$$ A \times B = \{(w, a), (w, b), (x, a), (x, b), (y, a), (y, b), (z, a), (z, b)\} $$
|
|
|
|
There are 8 elements in $A \times B$.
|
|
|
|
b. $B \times A$
|
|
|
|
$$ B \times A = \{(a, w), (b, w), (a, x), (b, x), (a, y), (b, y), (a, z), (b, z)\} $$
|
|
|
|
There are 8 elements in $B \times A$.
|
|
|
|
c. $A \times A$
|
|
|
|
$$ A \times A = \{(w, w), (w, x), (w, y), (w, z), (x, w), (x, x), (x, y), (x, z), (y, w), (y, x), (y, y), (y, z), (z, w), (z, x), (z, y), (z, z)\} $$
|
|
|
|
There are 16 elements in $A \times A$.
|
|
|
|
d. $B \times B$
|
|
|
|
$$ B \times B = \{(a, a), (a, b), (b, a), (b, b)\} $$
|
|
|
|
There are 4 elements in $B \times B$.
|
|
|
|
12. Let $S = \{2, 4, 6\}$ and $T = \{1, 3, 5\}$. Use the set-roster notation to
|
|
write each of the following sets, and indicate the number of elements that
|
|
are in each set.
|
|
|
|
a. $S \times T$
|
|
|
|
b. $T \times S$
|
|
|
|
c. $S \times S$
|
|
|
|
d. $T \times T$
|
|
|
|
**Solution**
|
|
|
|
a. $S \times T$
|
|
|
|
$$ S \times T = \{(2, 1), (2, 3), (2, 5), (4, 1), (4, 3), (4, 5), (6, 1), (6, 3), (6, 5)\} $$
|
|
|
|
There are 9 elements in $S \times T$.
|
|
|
|
b. $T \times S$
|
|
|
|
$$ T \times S = \{(1, 2), (1, 4), (1, 6), (3, 2), (3, 4), (3, 6), (5, 2), (5, 4), (5, 6)\} $$
|
|
|
|
There are 9 elements in $T \times S$.
|
|
|
|
c. $S \times S$
|
|
|
|
$$ S \times S = \{(2, 2), (2, 4), (2, 6), (4, 2), (4, 4), (4, 6), (6, 2), (6, 4), (6, 6)\} $$
|
|
|
|
There are 9 elements in $S \times S$.
|
|
|
|
d. $T \times T$
|
|
|
|
$$ T \times T = \{(1, 1), (1, 3), (1, 5), (3, 1), (3, 3), (3, 5), (5, 1), (5, 3), (5, 5)\} $$
|
|
|
|
There are 9 elements in $T \times T$.
|
|
|
|
13. Let $A = \{1, 2, 3\}$, $B = \{u\}$, and $C = \{m, n\}$. Find each of the
|
|
following sets.
|
|
|
|
a. $A \times (B \times C)$
|
|
|
|
b. $(A \times B) \times C$
|
|
|
|
c. $A \times B \times C$
|
|
|
|
**Solution**
|
|
|
|
a. $A \times (B \times C)$
|
|
|
|
$$ B \times C = \{(u, m), (u, n)\} $$
|
|
|
|
$$ A \times (B \times C) = \{(1, (u, m)), (1, (u, n)), (2, (u, m)), (2, (u, n)), (3, (u, m)), (3, (u, n))\} $$
|
|
|
|
b. $(A \times B) \times C$
|
|
|
|
$$ A \times B = \{(1, u), (2, u), (3, u)\} $$
|
|
|
|
$$ (A \times B) \times C = \{((1, u), m), ((1, u), n), ((2, u), m), ((2, u), n), ((3, u), m), ((3, u), n)\} $$
|
|
|
|
c. $A \times B \times C$
|
|
|
|
$$ A \times B \times C = \{(1, u, m), (1, u, n), (2, u, m), (2, u, n), (3, u, m), (3, u, n)\} $$
|
|
|
|
14. Let $R = \{a\}$, $S = \{x, y\}$, and $T = \{p, q, r\}$. Find each of the
|
|
following sets.
|
|
|
|
a. $R \times (S \times T)$
|
|
|
|
$$ S \times T = \{(x, p), (x, q), (x, r), (y, p), (y, q), (y, r)\} $$
|
|
|
|
$$ R \times (S \times T) = \{(a, (x, p)), (a, (x, q)), (a, (x, r)), (a, (y, p)), (a, (y, q)), (a, (y, r))\} $$
|
|
|
|
b. $(R \times S) \times T$
|
|
|
|
$$ R \times S = \{(a, x), (a, y)\} $$
|
|
|
|
$$ (R \times S) \times T = \{((a, x), p), ((a, x), q), ((a, x), r), ((a, y), p), ((a, y), q), ((a, y), r)\} $$
|
|
|
|
c. $R \times S \times T$
|
|
|
|
**Solution**
|
|
|
|
$$ R \times S \times T = \{(a, x, p), (a, x, q), (a, x, r), (a, y, p), (a, y, q), (a, y, r)\} $$
|
|
|
|
a. $R \times (S \times T)$
|
|
|
|
b. $(R \times S) \times T$
|
|
|
|
c. $R \times S \times T$
|
|
|
|
15. Let $S = \{0, 1\}$. List all the strings of length 4 over $S$ that contain
|
|
three or more $0$'s.
|
|
|
|
**Solution**
|
|
|
|
0000, 0001, 0010, 0100, 1000
|
|
|
|
16. Let $T = \{x, y\}$. List all the strings of length 5 over $T$ that have
|
|
exactly one $y$.
|
|
|
|
**Solution**
|
|
|
|
xxxxy, xxxyx, xxyxx, xyxxx, yxxxx
|
|
|
|
---
|
|
|
|
**Exercise Set 1.3**
|
|
|
|
Page 45
|
|
|
|
1. Let $A = \{2, 3, 4\}$ and $B = \{6, 8, 10\}$ and define a relation $R$ from
|
|
$A$ to $B$ as follows: For every $(x, y) \in A \times B$,
|
|
|
|
$$ (x, y) \in R \quad \text{ means that } \frac{y}{x} \text{ is an integer.} $$
|
|
|
|
**Solution**
|
|
|
|
a.
|
|
|
|
Is 4 _R_ 6?
|
|
|
|
No, $\dfrac{6}{4} = \dfrac{3}{2}$, which is not an integer.
|
|
|
|
Is 4 _R_ 8?
|
|
|
|
Yes, $\dfrac{8}{4} = 2$, which is an integer.
|
|
|
|
Is $(3, 8) \in R$?
|
|
|
|
No, $\dfrac{8}{3}$ is not an integer.
|
|
|
|
Is $(2, 10) \in R$?
|
|
|
|
Yes, $\dfrac{10}{2} = 5$ which is an integer.
|
|
|
|
b. Write _R_ as a set of ordered pairs.
|
|
|
|
$$ R = \{(2, 6), (2, 8), (2, 10), (3, 6), (4, 8)\} $$
|
|
|
|
c. Write the domain and co-domain of _R_.
|
|
|
|
The domain of _R_ is $\{2, 3, 4\}$.
|
|
|
|
The co-domain of _R_ is $\{6, 8, 10\}$.
|
|
|
|
d. Draw an arrow diagram for _R_.
|
|
|
|
2. Let $C = D = \{-3, -2, -1, 1, 2, 3\}$ and define a relation $S$ from $C$ to
|
|
$D$ as follows: For every $(x, y) \in C \times D$,
|
|
|
|
$$ (x, y) \in S \quad \text{ means that } \frac{1}{x} - \frac{1}{y} \text{ is an integer.} $$
|
|
|
|
**Solution**
|
|
|
|
a.
|
|
|
|
Is 2 _S_ 2?
|
|
|
|
Yes, $\dfrac{1}{2} - \dfrac{1}{2} = 0 \in \mathbb{Z}$.
|
|
|
|
Is -1 _S_ -1?
|
|
|
|
Yes $\dfrac{1}{-1} - \dfrac{1}{-1} = 0 \in \mathbb{Z} $
|
|
|
|
Is $(3, 3) \in S$?
|
|
|
|
Yes $\dfrac{1}{3} - \dfrac{1}{3} = 0 \in \mathbb{Z} $
|
|
|
|
Is $(3, -3) \in S$?
|
|
|
|
No, $\dfrac{1}{3} - \dfrac{1}{-3} = \dfrac{2}{3} \notin \mathbb{Z} $
|
|
|
|
b. Write _S_ as a set of ordered pairs.
|
|
|
|
$$ S = \{(-3, -3), (-2, -2), (-1, -1), (1, 1), (2, 2), (3, 3)\} $$
|
|
|
|
c. Write the domain and co-domain of _S_.
|
|
|
|
The domain and co-domain of _S_ is $\{-3, -2, -1, 1, 2, 3\}$.
|
|
|
|
d. Draw an arrow diagram for _S_.
|
|
|
|
3. Let $E = \{1, 2, 3\}$ and $F = \{-2, -1, 0\}$ and define a relation $T$ from
|
|
$E$ to $F$ as follows: For every $(x, y) \in E \times F$,
|
|
|
|
$$ (x, y) \in T \quad \text{ means that } \frac{x - y}{3} \text{ is an integer.} $$
|
|
|
|
**Solution**
|
|
|
|
a.
|
|
|
|
Is 3 _T_ 0?
|
|
|
|
Yes, $\dfrac{3 - 0}{3} = 1 \in \mathbb{Z}$.
|
|
|
|
Is 1 _T_ (-1)?
|
|
|
|
No, $\dfrac{(1) - (-1)}{3} = \dfrac{2}{3} \notin \mathbb{Z}$.
|
|
|
|
Is $(2, -1) \in T$?
|
|
|
|
Yes, $\dfrac{(2) - (-1)}{3} = 1 \in \mathbb{Z}$.
|
|
|
|
Is $(3, -2) \in T$?
|
|
|
|
No, $\dfrac{(3) - (-2)}{3} = \dfrac{5}{3} \notin \mathbb{Z}$.
|
|
|
|
b. Write $T$ as a set of ordered pairs.
|
|
|
|
$$ T = \{(1, -2), (2, -1), (3, 0)\} $$
|
|
|
|
c. Write the domain and co-domain of $T$.
|
|
|
|
The domain of $T$ is $\{1, 2, 3\}$, and the co-domain of $T$ is $\{-2, -1, 0\}$.
|
|
|
|
d. Draw an arrow diagram for $T$.
|
|
|
|
4. Let $G = \{-2, 0, 2\}$ and $H = \{4, 6< 8\}$ and define a relation $V$ from
|
|
$G$ to $H$ as follows: For every $(x, y) \in G \times H$,
|
|
|
|
$$ (x, y) \in V \quad \text{ means that } \frac{x - y}{4} \text{ is an integer.} $$
|
|
|
|
**Solution**
|
|
|
|
a.
|
|
|
|
Is 2 _V_ 6?
|
|
|
|
Yes, $\dfrac{(2) - (6)}{4} = -1 \in \mathbb{Z}$.
|
|
|
|
Is (-2) _V_ (8)?
|
|
|
|
No, $\dfrac{(-2) - (8)}{4} = -\dfrac{10}{4} = -\dfrac{5}{2} \notin \mathbb{Z}$.
|
|
|
|
Is $(0, 6) \in V$?
|
|
|
|
No, $\dfrac{(0) - (6)}{4} = -\dfrac{6}{4} = -\dfrac{3}{2} \notin \mathbb{Z}$.
|
|
|
|
Is $(2, 4) \in V$?
|
|
|
|
No, $\dfrac{(2) - (4)}{4} = -\dfrac{1}{2} \notin \mathbb{Z}$.
|
|
|
|
b. Write $V$ as a set of ordered pairs.
|
|
|
|
$$ V = \{(-2, 6), (0, 4), (0, 8), (2, 6)\} $$
|
|
|
|
c. Write the domain and co-domain of _V_.
|
|
|
|
The domain of _V_ is $\{-2, 0, 2\}$ and the co-domain of _V_ is $\{4, 6, 8\}$
|
|
|
|
d. Draw an arrow diagram for _V_.
|
|
|
|
5. Define a relation $S$ from $\mathbb{R}$ to $\mathbb{R}$ as follows: For every
|
|
$(x, y) \in \mathbb{R} \times \mathbb{R}$,
|
|
|
|
$$ (x, y) \in S \quad \text{ means that } x \geq y $$
|
|
|
|
**Solution**
|
|
|
|
a.
|
|
|
|
Is $(2, 1) \in S$?
|
|
|
|
Yes, $(2) \geq (1)$.
|
|
|
|
Is $(2, 2) \in S$?
|
|
|
|
Yes, $(2) \geq (2)$.
|
|
|
|
Is 2 _S_ 3?
|
|
|
|
No, $(2) \cancel{\geq} (3)$.
|
|
|
|
Is (-1) _S_ (-2)?
|
|
|
|
Yes, $(-1) \geq (-2)$.
|
|
|
|
b. Draw the graph of _S_ in the Cartesian plane.
|
|
|
|
6. Define a relation $R$ from $\mathbb{R}$ to $\mathbb{R}$ as follows: For every
|
|
$(x, y) \in \mathbb{R} \times \mathbb{R}$,
|
|
|
|
$$ (x, y) \in R \quad \text{ means that } y = x^2 $$
|
|
|
|
**Solution**
|
|
|
|
a.
|
|
|
|
Is $(2, 4) \in R$?
|
|
|
|
Yes, $(4) = (2)^2$.
|
|
|
|
Is $(4, 2) \in R$?
|
|
|
|
No, $(2) \neq (4)^2$.
|
|
|
|
Is (-3) _R_ 9?
|
|
|
|
Yes, $(9) = (-3)^2$.
|
|
|
|
Is 9 _R_ (-3)?
|
|
|
|
No, $(-3) \neq (9)^2$.
|
|
|
|
b. Draw the graph of _R_ in the Cartesian plane.
|
|
|
|
7. Let $A = \{4, 5, 6\}$ and $B = \{5, 6, 7\}$ and define relations $R$, $S$,
|
|
and $T$ from $A$ to $B$ as follows: For every $(x, y) \in A \times B$:
|
|
|
|
$$ (x, y) \in R \quad \text{ means that } x \geq y $$
|
|
|
|
$$ (x, y) \in S \quad \text{ means that } \frac{x - y}{2} \text{ is an integer.} $$
|
|
|
|
$$ T = \{(4, 7), (6, 5), (6, 7)\} $$
|
|
|
|
**Solution**
|
|
|
|
a. Draw arrow diagrams for $R$, $S$, and $T$.
|
|
|
|
b. Indicate whether any of the relations $R$, $S$, and $T$ are functions.
|
|
|
|
$R$ is not a function because it satisfies neither property (1) nor property (2)
|
|
of the definition. It fails property (1) because $(4, y) \not in R$, for any $y$
|
|
in $B$. It fails property (2) because $(6, 5) \in R$ and $(6, 6) \in R$ and
|
|
$5 \neq 6$.
|
|
|
|
$S$ is not a function because $(5, 5) \in S$ and $(5, 7) \in S$ and $5 \neq 7$.
|
|
So $S$ does not satisfy property (2) of the definition of a function.
|
|
|
|
$T$ is not a function both because $(5, x) \notin T$ for any $x$ in $B$ and
|
|
because $(6, 5) \in T$ and $(6, 7) \in T$ and $5 \neq 7$. So $T$ does not
|
|
satisfy either property (1) or property (2) of the definition of a function.
|
|
|
|
8. Let $A = \{2, 4\}$ and $B = \{1, 3, 5\}$ and define relations $U$, $V$, and
|
|
$W$ from $A$ to $B$ as follows: For every $(x, y) \in A \times B$:
|
|
|
|
$$ (x, y) \in U \quad \text{ means that } y - x > 2 $$
|
|
|
|
$$ (x, y) \in V \quad \text{ means that } y - 1 = \frac{x}{2} $$
|
|
|
|
W = \{(2, 5), (4, 1), (2, 3)\}
|
|
|
|
**Solution**
|
|
|
|
a. Draw arrow diagrams for $U$, $V$, and $W$.
|
|
|
|
b. Indicate whether any of the relations $U$, $V$, and $W$ are functions.
|
|
|
|
$U$ is not a function by property (1), as $(4, y) \notin B$.
|
|
|
|
$V$ is not a function by property (1) as $(2, y) \notin B$.
|
|
|
|
$T$ is not a function by property (2) as $(2, 3) \in B$ and $(2, 5) \in B$ and
|
|
$3 \neq 5$.
|
|
|
|
9.
|
|
|
|
**Solution**
|
|
|
|
a. Find all functions from $\{0, 1\}$ to $\{1\}$.
|
|
|
|
$$ \{(0, 1), (1, 1)\} $$
|
|
|
|
b. Find two relations form $\{0, 1\}$ to $\{1\}$ that are not functions.
|
|
|
|
$$ \{(0, 1)\}, \{(1, 1)\} $$
|
|
|
|
10. Find four relations from $\{a, b\}$ to $\{x, y\}$ that are not functions
|
|
from $\{a, b\}$ to $\{x, y\}$.
|
|
|
|
**Solution**
|
|
|
|
$$ \{(a, x)\}, \{(a, y)\}, \{(b, x)\}, \{(b, y)\} $$
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11. Let $A = \{0, 1, 2\}$ and let $S$ be the set of all strings over $A$. Define
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a relation $L$ from $S$ to $\mathbb{Z}^{\text{nonneg}}$ as follows: For
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every string $s$ in $S$ and every nonnegative integer $n$,
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$$ (s, n) \in L \quad \text{ means that the length of } s \text{ is } n $$
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Then $L$ is a function because every string in $S$ has one and only one length.
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Find $L(0201)$ and $L(12)$.
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**Solution**
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$$ L(0201) = 4 $$
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$$ L(12) = 2 $$
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12. Let $A = \{x, y\}$ and let $S$ be the set of all strings over $A$. Define a
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relation $C$ from $S$ to $S$ as follows: For all strings $s$ and $t$ in $S$,
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$$ (s, t) \in C \quad \text{ means that } t = ys $$
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Then $C$ is a function because every string in $S$ consists entirely of $x$'s
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and $y$'s and adding an additional $y$ on the left creates a single new string
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that consists of $x$'s and $y$'s and is, therefore, also in $S$. Find $C(x)$ and
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$C(yyxyx)$.
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**Solution**
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$$ C(x) = yx $$
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$$ C(yyxyx) = yyyxyx $$
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13. Let $A = \{-1, 0, 1\}$ and $B = \{t, u, v, w\}$. Define a function
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$F: A \to B$ by the following arrow diagram:
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**Solution**
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a. Write the domain and co-domain of $F$.
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The domain of $F$ is $\{-1, 0, 1\}$, and the co-domain of $F$ is
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$\{t, u, v, w\}$.
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b. Find $F(-1)$, $F(0)$, and $F(1)$.
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$$ F(-1) = u $$
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$$ F(0) = w $$
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$$ F(1) = u $$
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14. Let $C = \{1, 2, 3, 4\}$ and $D = \{a, b, c, d\}$. Define a function
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$G: C \to D$ by the following diagram:
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**Solution**
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a. Write the domain and co-domain of $G$.
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The domain of $G$ is $\{1, 2, 3, 4\}$, and the co-domain of $G$ is
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$\{a, b, c, d\}$.
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b. Find $G(1)$, $G(2)$, $G(3)$, and $G(4)$.
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$$ G(1) = c $$
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$$ G(2) = c $$
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$$ G(3) = c $$
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$$ G(4) = c $$
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15. Let $X = \{2, 4, 5\}$ and $Y = \{1, 2, 4, 6\}$. Which of the following arrow
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diagrams determine functions from $X$ to $Y$?
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**Solution**
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Only (d) is a function.
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(a) is not a function by property (2), as $(2, 1) \in X \to Y$ and
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$(2, 6) \in X \to Y$, and $1 \neq 6$.
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(b) is not a function by property (1), as $(5, y) \notin X \to Y$.
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(c) is not a function by property (2), as $(4, 1) \in X \to Y$ and
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$(4, 2) \in X \to Y$ and $(1 \neq 2)$.
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(e) is not a function by property (1), as $(2, y) \notin X \to Y$.
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16. Let $f$ be the squaring function defined in Example 1.3.6. Find $f(-1)$,
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$f(0)$, and $f\left(\dfrac{1}{2}\right)$.
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**Solution**
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$$ f(-1) = (-1)^2 = 1 $$
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$$ f(0) = (0)^2 = 0 $$
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$$ f\left(\frac{1}{2}\right) = \left(\frac{1}{2}\right)^2 = \frac{1}{4} $$
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17. Let $g$ be the successor function defined in Example 1.3.6. Find $g(-1000)$,
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$g(0)$, and $g(999)$.
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**Solution**
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$$ g(-1000) = (-1000) + 1 = -999 $$
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$$ g(0) = (0) + 1 = 1 $$
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$$ g(999) = (999) + 1 = 1000 $$
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18. Let $h$ be the constant function defined in Example 1.3.6. Find
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$h\left(-\dfrac{12}{5}\right)$, $h\left(\dfrac{0}{1}\right)$, and
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$h\left(\dfrac{9}{17}\right)$.
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**Solution**
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$$ h\left(-\frac{12}{5}\right) = 2 $$
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$$ h\left(\frac{0}{1}\right) = 2 $$
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$$ h\left(\frac{9}{17}\right) = 2 $$
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19. Define functions $f$ and $g$ from $\mathbb{R}$ to $\mathbb{R}$ by the
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following formulas: For every $x \in \mathbb{R}$,
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$$ f(x) = 2x \quad \text{ and } \quad g(x) = \frac{2x^3 + 2x}{x^2 + 1} $$
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Does $f = g$? Explain.
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**Solution**
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Yes, by factoring out $2x$ from the numerator of $g(x)$ we find they are the
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same function:
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$$ g(x) = \frac{2x^3 + 2x}{x^2 + 1} = \frac{2x(x^2 + 1)}{(x^2 + 1)} = 2x = f(x) $$
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This means that for every input $x$ to both $g$ and $f$, $f(x) = g(x)$, and so
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$f = g$ by definition of equality of functions.
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20. Define functions $H$ and $K$ from $\mathbb{R}$ to $\mathbb{R}$ by the
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following formulas: For every $x \in \mathbb{R}$,
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$$ H(x) = (x - 2)^2 \quad \text{ and } \quad K(x) = (x - 1)(x - 3) + 1 $$
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Does $H = K$? Explain.
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**Solution**
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$$ H(x) = (x - 2)^2 = (x - 2)(x - 2) = x^2 - 4x + 4 $$
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$$ K(x) = (x - 1)(x - 3) + 1 = x^2 - 4x + 3 + 1 = x^2 - 4x + 4 $$
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Therefore $H(x) = K(x)$ by the definition of equality of functions.
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