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🚧 Progress through chapter 1

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**Example 1.1.1**
Page 24
Use variables to rewrite the following sentences more formally.
a. Are there numbers with the property that the sum of their squares equals the
square of their sum?
b. Given any real number, its square is nonnegative.
**Solution**
a. Are there numbers $a$ and $b$ with the property that $a^2 + b^2 = (a + b)^2$?
_Or_: Are there numbers $a$$ and $b$ such that $a^2 + b^2 = (a + b)^2$?
_Or_: Do there exist any numbers $a$ and $b$ such that $a^2 + b^2 = (a + b)^2$?
b. Given any real number $r$, $r^2$ is nonnegative.
_Or_: For any real number $r$, $r^2 \geq 0$.
_Or_: For every real number $r$, $r^2 \geq 0$.
---
**Example 1.1.2**
Page 26
Fill in the blanks to rewrite the following statement:
For every real number $x$, if $x$ is nonzero then $x^2$ is positive.
a. If a real number is nonzero, then its square ________.
b. For every nonzero real number $x$, ________.
c. If $x$ ________, then ________.
d. The square of any nonzero real number is ________.
e. All nonzero real numbers have ________.
**Solution**.
a. is positive.
b. $x^2$ is positive.
c. is a nonzero real number, $x^2$ is positive.
d. positive.
e. positive squares .
---
**Example 1.1.3**
Page 27
Fill in the blanks to rewrite the following statement: Every pot has a lid.
a. All pots ________.
b. For every pot $P$, there is ________.
c. For every pot $P$, there is a lid $L$ such that ________.
**Solution**
a. have lids.
b. a lid.
c. $L$ is a lid for $P$..
---
**Example 1.1.4**
Page 28
Fill in the blanks to rewrite the following statement in three different ways:
There is a person in my class who is at least as old as every person in my clas.
a. Some ________ is at least as old as ________.
b. There is a person $p$ in my class such that $p$ is ________.
c. There is a person $p$ in my class with the property that for every person $q$
in my class, $p$ is ________.
**Solution**
a. person; every person.
b. at least as old as every person in my class.
c. at least as old as $q$.
---
**Example 1.2.1**
Page 30
**Using the Set-Roster Notation**
a. Let $A = \{1, 2, 3\}$, $B = \{3, 1, 2\}$, and $C = \{1, 1, 2, 3, 3, 3\}$.
What are the elements of $A$, $B$, and $C$? How are $A$, $B$, and $C$ related?
b. Is $\{0\} = 0$?
c. How many elements are in the set $\{1, \{1}\}$?
d. For each nonnegative integer $n$, let $U_n = \{n, -n\}$. Find $U_1$, $U_2$,
and $U_0$.
**Solution**
a. Let $A = \{1, 2, 3\}$, $B = \{3, 1, 2\}$, and $C = \{1, 1, 2, 3, 3, 3\}$.
What are the elements of $A$, $B$, and $C$? How are $A$, $B$, and $C$ related?
$A$, $B$, and $C$ have exactly the same three elements, $1$, $2$, and $3$.
Therefore, $A$, $B$, and $C$ are simply different ways to represent the same
set.
b. Is $\{0\} = 0$?
$\{0\} \neq 0$ because $\{0\}$ is a set with one element, namely $0$, whereas
$0$ is just the symbol that represents the number zero.
c. How many elements are in the set $\{1, \{1}\}$?
The set $\{1, \{1\}\}$ has two elements. $1$ and the set whose only element is
$1$.
d. For each nonnegative integer $n$, let $U_n = \{n, -n\}$. Find $U_1$, $U_2$,
and $U_0$.
$U_1 = \{1, -1\}, \quad U_2 = \{2, -2\}, \quad U_0 = \{0, 0\} = \{0\}$
---
**Example 1.2.2**
Page 31
**Using the Set-Builder Notation**
Given that $\mathbb{R}$ denotes the set of all real numbers, $\mathbb{Z}$ the
set of all integers, and $\mathbb{Z}^+$ the set of all positive integers,
describe each of the following sets.
a. $\{x \in \mathbb{R} | -2 < x < 5\}$
b. $\{x \in \mathbb{Z} | -2 < x < 5\}$
c. $\{x \in \mathbb{Z}^+ | -2 < x < 5\}$
**Solution**
a. $\{x \in \mathbb{R} | -2 < x < 5\}$
$\{x \in \mathbb{R} | -2 < x < 5\}$ is the open interval of real numbers
(strictly) between $-2$ and 5. It is pictured as follows (see page 31).
b. $\{x \in \mathbb{Z} | -2 < x < 5\}$
$\{x \in \mathbb{Z} | -2 < x < 5\}$ is the set of all integers (strictly)
between $-2$ and $5$. It is equal to the set $\{-1, 0, 1, 2, 3, 4}$.
c. $\{x \in \mathbb{Z}^+ | -2 < x < 5\}$
Since all the integers in $\mathbb{Z}^+$ are positive,
$\{x \in \mathbb{Z}^+ | -2 < x < 5\} = \{1, 2, 3, 4\}$.
---
**Example 1.2.3**
Page 32
Let $A = \mathbb{Z}^+$, $B = \{n \in \mathbb{Z} | 0 \leq n \leq 100\}$, and
$C = \{100, 200, 300, 400, 500\}$. Evaluate the truth and falsity of each of the
following statements
a. $B \subseteq A$
b. $C$ is a proper subset of $A$.
c. $C$ and $B$ have at least one element in common
d. $C \subseteq B$
e. $C \subseteq C$
**Solution**
a. $B \subseteq A$
False. Zero is not a positive integer. Thus zero is in $B$ but zero is not in
$A$, and so $B \nsubseteq A$
b. $C$ is a proper subset of $A$.
True. Each element in $C$ is a positive integer, and hence, is in $A$, but there
are elements in $A$ that are not in $C$. For instance, $1$ is in $A$ and not in
$C$.
c. $C$ and $B$ have at least one element in common
True. For example, $100$ is in both $C$ and $B$.
d. $C \subseteq B$
False. For example, $200$ is in $C$ but not in $B$.
e. $C \subseteq C$
True. Every element in $C$ is in $C$. In general, the definition of a subset
implies that all sets are subsets of themselves.
---
**Example 1.2.4**
Page 33
**Distinction between $\in$ and $\subseteq$**
Which of the following are true statements?
a. $2 \in \{1, 2, 3\}$
b. $\{2\} \in \{1, 2, 3\}$
c. $2 \subseteq \{1, 2, 3\}$
d. $\{2\} \subseteq \{1, 2, 3\}$
e. $\{2\} \subseteq \{\{1\}, \{2\}\}$
f. $\{2} \in \{\{1\}, \{2\}\}$
**Solution**
Only (a), (d), and (f) are true.
For (b) to be true, the set $\{1, 2, 3\}$ would have to contain the element
$\{2\}$. But the only elements of $\{1, 2, 3\}$ are $1$, $2$, and $3$, and $2$
is not equal to $\{2\}$. Hence (b) is false.
For \(c\) to be true, the number $2$ would have to be a set and every element in
the set $2$ would have to be an element of $\{1, 2, 3}$. This is not the case,
so \(c\) is false.
For (e) to be true, every element in the set containing only the number $2$
would have to be an element of the set whose elements are $\{1\}$ and $\{2\}$.
But $2$ is not equal to either $\{1\}$ or $\{2\}$, and so (e) is false.
---
**Example 1.2.5 Ordered Pairs**
Page 34
a. Is $(1, 2) = (2, 1)$?
b. Is $\left(3, \dfrac{5}{10}\right) = \left(\sqrt{9}, \dfrac{1}{2}\right)$?
c. What is the first element of $(1, 1)$?
**Solution**
a. Is $(1, 2) = (2, 1)$?
No, By definition of equality of ordered pairs,
$(1, 2) = (2, 1)$ if, and only if, 1 = 2, and 2 = 1.
But $1 \neq 2$, and so the ordered pairs are not equal.
b. Is $\left(3, \dfrac{5}{10}\right) = \left(\sqrt{9}, \dfrac{1}{2}\right)$?
Yes. By definition of equality of ordered pairs,
$\left(3, \dfrac{5}{10}\right) = \left(\sqrt{9}, \dfrac{1}{2}\right)$ if, and
only if, $3 = \sqrt{9}$ and $\dfrac{5}{10} = \dfrac{1}{2}$.
Because these equations are both true, the ordered pairs are equal.
c. What is the first element of $(1, 1)$?
In the ordered pair $(1, 1)$, the first and second elements are both $1$.
---
**Example 1.2.6 Ordered $n$-tuples**
Page 34
a. Is $(1, 2, 3, 4) = (1, 2, 4, 3)$?
b. Is
$\left(3, (-2)^2, \dfrac{1}{2}\right) = \left(\sqrt{9}, 4, \dfrac{3}{6}\right)$?
**Solution**
a. Is $(1, 2, 3, 4) = (1, 2, 4, 3)$?
No. By definition of equality of ordered 4-tuples,
$$ (1, 2, 3, 4) = (1, 2, 4, 3) \leftrightarrow 1 = 1, 2 = 2, 3 = 4, and 4 = 3 $$
But $3 \neq 4$, and so the ordered 4-tuples are not equal.
b. Is
$\left(3, (-2)^2, \dfrac{1}{2}\right) = \left(\sqrt{9}, 4, \dfrac{3}{6}\right)$?
Yes. By definition of equality of ordered triples.
$$ \left(3, (-2)^2, \frac{1}{2}\right) = \left(\sqrt{9}, 4, \frac{3}{6}\right) \leftrightarrow 3 = \sqrt{9} \text{ and } (-2)^2 = 4 \text{ and } \frac{1}{2} = \frac{3}{6} $$
Because these equations are all true, the two ordered triples are equal.
---
**Example 1.2.7 Cartesian Products**
Page 35
Let $A = \{x, y\}$, $B = \{1, 2, 3\}$, and $C = \{a, b\}$.
a. Find $A \times B$.
b. Find $B \times A$.
c. Find $A \times A$.
d. How many elements are in $A \times B$, $B \times A$, and $A \times A$?
e. Find $(A \times B) \times C$
f. Find $A \times B \times C$
g. Let $\mathbb{R}$ denote the set of all real numbers. Describe
$\mathbb{R} times \mathbb{R}$.
**Solution**
a. Find $A \times B$.
$$ A \times B = \{(x, 1), (y, 1), (x, 2), (y, 2), (x, 3), (y, 3)\} $$
b. Find $B \times A$.
$$ B \times A = \{(1, x), (1, y), (2, x), (2, y), (3, x), (3, y)\} $$
c. Find $A \times A$.
$$ A \times A = \{(x, x), (x, y), (y, x), (y, y)\} $$
d. How many elements are in $A \times B$, $B \times A$, and $A \times A$?
$A \times B$ has 6 elements. Note that this is the number of elements in $A$
times the number of elements in $B$. $B \times A$ has 6 elements, the number of
elements in $B$ times the number of elements in $A$. $A \times A$ has 4
elements, the number of elements in $A$ times the number of elements in $A$.
e. Find $(A \times B) \times C$
$$ (A \times B) \times C = \{(u, v) | u \in A \times B \text{ and } v \in C\} $$
By definition of Cartesian product.
$$ (A \times B) \times C = \{((x, 1), a), ((x, 2), a), ((x, 3), a), ((y, 1), a), ((y, 2), a), ((y, 3), a), ((x, 1), b), ((x, 2), b), ((x, 3), b), ((y, 1), b), ((y, 2), b), ((y, 3), b)\} $$
f. Find $A \times B \times C$
The Cartesian product $A \times B \times C$ is superficially similar to but is
not quite the same mathematical object as $(A \times B) \times C$.
$(A \times B) \times C$ is a set of ordered pairs of which one element is itself
an ordered pair, whereas $A \times B \times C$ is a set of ordered triples. By
definition of Cartesian product,
$$ A \times B \times C = \{(u, v, w) | u \in A, v \in B, \text{ and } w \in C\} $$
$$ A \times B \times C = \{(x, 1, a), (x, 2, a), (x, 3, a), (y, 1, a), (y, 2, a), (y, 3, a), (x, 1, b), (x, 2, b), (x, 3, b), (y, 1, b), (y, 2, b), (y, 3, b)\} $$
g. Let $\mathbb{R}$ denote the set of all real numbers. Describe
$\mathbb{R} times \mathbb{R}$.
$\mathbb{R} \times \mathbb{R}$ is the set of all ordered pairs $(x, y)$ where
both $x$ and $y$ are real numbers. If horizontal and vertical axes are drawn on
a plane and a unit length is marked off, then each ordered pair in
$\mathbb{R} \times \mathbb{R}$ corresponds to a unique point in the plane, with
the first and second elements o the pair indicating, respectively, the
horizontal and vertical positions of the point. The term **Cartesian plane** is
often used to refer to a plane with this coordinate system, as illustrated in
Figure 1.2.1 (see page 36).
---
**Example 1.2.8 Strings**
Page 36
Let $A = \{a, b\}$. List all the strings of length 3 over $A$ with at least two
characters that are the same.
**Solution**
_aab, aba, baa, aaa, bba, bab, abb, bbb_
In computer programming it is important to distinguish among different kinds of
data structures and to respect the notations that are used for them. Similarly
in mathematics, it is important to distinguish among, say, _{a, b, c}, {{ab},
c}, (a, b, c), (a, (b, c)), abc_ and so forth, because these are all
significantly different objects.

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**Exercise Set 1.1**
Page 28
In each of 1-6, fill in the blanks using a variable or variables to rewrite the
given statement.
1. Is there a real number whose square is $-1$?
a. Is there a real number $x$ such that ______?
b. Does there exist ______ such that $x^2 = -1$?
**Solution**
a. Is there a real number $x$ such that $x^2 = -1$?
b. Does there exist a real number $x$ such that $x^2 = -1$?
2. Is there an integer that has a remainder of $2$ when it is divided by $5$ and
a remainder of $3$ when it is divided by $6$?
a. Is there an integer $n$ such that $n$ has ______?
b. Does there exist ______ such that if $n$ is divided by $5$ the remainder is
$2$ and if ______?
_Note: There are integers with this property. Can you think of one?_
**Solution**
a. Is there an integer $n$ such that $n$ has a remainder of $2$ when $n$ is
divided by $5$ and a remainder of $3$ when $n$ is divided by $6$?
b. Does there exist a number $n$ such that if $n$ is divided by $5$ the
remainder is $2$ and if $n$ is divided by $6$ the remainder is $3$?
_Note: There are integers with this property. Can you think of one?_
$$ 27 \mod 5 = 2 $$
$$ 27 \mod 6 = 3 $$
3. Given any two distinct real numbers, there is a real number in between them.
a. Given any two distinct real numbers $a$ and $b$, there is a real number $c$
such that $c$ is ______.
b. For any two ______, ______ such that $c$ is between $a$ and $b$.
**Solution**
a. Given any two distinct real numbers $a$ and $b$, there is a real number $c$
such that $c$ is $a \leq c \leq b$.
b. For any two distinct real numbers $a$ and $b$, there exists a real number $c$
such that $c$ is between $a$ and $b$.
4. Given any real number, there is a real number that is greater.
a. Given any real number $r$, there is ______ $s$ such that $s$ is ______
b. For any ______, ______ such that $s > r$.
**Solution**
a. Given any real number $r$, there is a real number $s$ such that $s$ is
greater than $r$.
b. For any real number $r$, there exists a real number $s$ such that $s > r$.
5. The reciprocal of any positive real number is positive.
a. Given any positive real number $r$, the reciprocal of ______.
b. For any real number $$, if $r$ is ______, then ______.
c. If a real number $r$ ______, then ______.
**Solution**
a. Given any positive real number $r$, the reciprocal of $r$ is positive.
b. For any real number $r$, if $r$ is positive, then the reciprocal of $r$ is
positive.
c. If a real number $r$ is positive, then the reciprocal of $r$ is positive.
6. The cube root of any negative real number is negative.
a. Given any negative real number $s$, the cube root of ______.
b. For any real number $s$, if $s$ is ______, then ______.
c. If a real number $s$ ______, then ______.
**Solution**
a. Given any negative real number $s$, the cube root of $s$ is negative.
b. For any real number $s$, if $s$ is negative, then the cube root of $s$ is
negative.
c. If a real number $s$ is negative, then the cube root of $s$ is negative.
7. Rewrite the following statements less formally, without using variables.
Determine, as best as you can, whether the statements are true or false.
a. There are real numbers $u$ and $v$ with the property that $u + v < u - v$.
b. There is a real number $x$ such that $x^2 < x$.
c. For every positive integer $n$, $n^2 \geq n$.
d. For all real numbers $a$ and $b$, $|a + b| \leq |a| + |b|$.
**Solution**
a. There are real numbers $u$ and $v$ with the property that $u + v < u - v$.
There are two distinct real numbers where the sum of those two numbers is less
than the difference of those two numbers.
This is true if you consider our domain is all real numbers which include
negatives. For example:
$$ 1 + (-1) = 0 $$
$$ 1 - (-1) = 2 $$
$$ 0 < 2 $$
b. There is a real number $x$ such that $x^2 < x$.
There is a real number which is greater than it's square.
This is true for any fraction/decimal. Consider:
$$ \left(\frac{1}{4}\right)^2 = \frac{1}{16} $$
$$ \frac{1}{16} < \frac{1}{4} $$
c. For every positive integer $n$, $n^2 \geq n$.
For all positive integers, an integer's square is always greater than or equal
to the integer.
This is true. Starting at $1$ we get $1^2 \geq 1$, which is true, $2^2 \geq 2$
is true, and so on. We're essentially multiplying each side of the inequality by
some positive integer, which we know from algebra does not change the direction
of the inequality, so this statement holds true.
d. For all real numbers $a$ and $b$, $|a + b| \leq |a| + |b|$.
For any two distinct real numbers, the absolute value of their sum is less than
or equal to the sum of the absolute values of each number.
This is true, if both $a$ and $b$ are positive numbers or both $a$ and $b$ are
negative integers, then the two statements are equal. If either $a$ or $b$ is
negative and the other is positive, then the left statement will always be less
than the right hand statement.
---
In each of 8-13, fill in the blanks to rewrite the given statement.
8. For every object $J$, if $J$ is a square then $J$ has four sides.
a. All squares ______.
b. Every square ______.
c. If an object is a square, then it ______.
d. If $J$ ______, then $J$ ______.
e. For every square $J$, ______.
**Solution**
a. All squares have four sides.
b. Every square has four sides.
c. If an object is a square, then it has four sides.
d. If $J$ is a square, then $J$ has four sides.
e. For every square $J$, $J$ has four sides.
9. For every equation $E$, if $E$ is quadratic then $E$ has at most two real
solutions.
a. All quadratic equations ______.
b. Every quadratic equation ______.
c. If an equation is quadratic, then it ______.
d. If $E$ ______, then $E$ ______.
e. For every quadratic equation $E$, ______.
**Solution**
a. All quadratic equations have at most two real solutions.
b. Every quadratic equation has at most two real solutions.
c. If an equation is quadratic, then it has at most two real solutions.
d. If $E$ is a quadratic equation, then $E$ has at most two real solutions.
e. For every quadratic equation $E$, $E$ has at most two real solutions.
10. Every nonzero real number has a reciprocal.
a. All nonzero real numbers ______.
b. For every nonzero real number $r$, there is ______ for $r$.
c. For every nonzero real number $r$, there is a real number $s$ such that
______.
**Solution**
a. All nonzero real numbers have reciprocals.
b. For every nonzero real number $r$, there is a reciprocal for $r$.
c. For every nonzero real number $r$, there is a real number $s$ such that $s$
is a reciprocal of $r$.
11. Every positive number has a positive square root.
a. All positive numbers ______.
b. For every positive number $e$, there is ______ for $e$.
c. For every positive number $e$, there is a positive number $r$ such that
______.
**Solution**
a. All positive numbers have positive square roots.
b. For every positive number $e$, there is a positive square root for $e$.
c. For every positive number $e$, there is a positive number $r$ such that $r$
is a positive square root for $e$.
12. There is a real number whose product with every number leaves the number
unchanged.
a. Some ______ has the property that its ______.
b. There is a real number $r$ such that the product of $r$ ______.
c. There is a real number $r$ with the property that for every real number $s$,
______.
**Solution**
a. Some real number has the property that its product with every number leaves
the number unchanged.
b. There is a real number $r$ such that the product of $r$ with every number
leaves $r$ unchanged.
c. There is a real number $r$ with the property that for every real number $s$,
such that $rs = s$.
13. There is a real number whose product with every real number equals zero.
a. Some _____ has the property that its ______.
b. There is a real number $a$ such that the product of $a$ ______.
c. There is a real number $a$ with the property that for every real number $b$,
______.
**Solution**
a. Some real number has the property that its product with every real number
equals zero.
b. There is a real number $a$ such that the product of $a$ with every real
number equals zero.
c. There is a real number $a$ with the property that for every real number $b$,
$ab = 0$.
---
**Exercise Set 1.2**
Page 37
1. Which of the following sets are equal?
$$ A = \{a, b, c, d\} $$
$$ B = \{d, e, a, c\} $$
$$ C = \{d, b, a, c\} $$
$$ D = \{a, a, d, e, c, e\} $$
**Solution**
$$ A = C $$
$$ B = D $$
2. Write in words how to read each of the following out loud.
a. $\{x \in \mathbb{R}^+ | 0 < x < 1\}$
b. $\{x \in \mathbb{R} | x \leq 0 \text{ or } x \geq 1\}$
c. $\{n \in \mathbb{Z} | n \text{ is a factor of } 6\}$
d. $\{n \in \mathbb{Z}^+ | n \text{ is a factor of } 6\}$
**Solution**
a. $\{x \in \mathbb{R}^+ | 0 < x < 1\}$
The set of all positive real numbers $x$ such that $x$ is greater than $0$ and
less than $1$.
b. $\{x \in \mathbb{R} | x \leq 0 \text{ or } x \geq 1\}$
The set of all real numbers $x$ such that $x$ is less than or equal to $0$ or
$x$ is greater than or equal to $1$.
c. $\{n \in \mathbb{Z} | n \text{ is a factor of } 6\}$
The set of all integers $n$ such that $n$ is a factor of $6$.
d. $\{n \in \mathbb{Z}^+ | n \text{ is a factor of } 6\}$
The set of all positive integers $n$ such that $n$ is a factor of $6$.
3.
a. Is $4 = \{4\}$?
b. How many elements are in the set $\{3, 4, 3, 5\}$?
c. How many elements are in the set $\{1, \{1\}, \{1, \{1\}\}\}$ ?
**Solution**
a. Is $4 = \{4\}$?
No, the symbol $4$, which represents the number four, does not equal the set
that contains an element that is the number $4$.
b. How many elements are in the set $\{3, 4, 3, 5\}$?
There are 3 elements in the set $\{3, 4, 3, 5\}$. Repeated elements are not
counted as more than 1 element in a set.
c. How many elements are in the set $\{1, \{1\}, \{1, \{1\}\}\}$ ?
There are three elements in the set, namely the symbol $1$, the set $\{1\}$, and
the set $\{1, \{1\}\}$.
4.
a. Is $2 \in \{2}$ ?
b. How many elements are in the set $\{2, 2, 2, 2\}$ ?
c. How many elements are in the set $\{0, \{0\}\}$ ?
d. Is $\{0\} \in \{\{0\}, \{1\}\}$ ?
e. Is $0 \in \{\{0\}, \{1\}\}$ ?
**Solution**
a. Is $2 \in \{2}$ ?
No, the symbol $2$ which represents the number two, is not equal to the set
$\{2\}$, which is a set that contains the element $2$.
b. How many elements are in the set $\{2, 2, 2, 2\}$ ?
There is one element in the set $\{2, 2, 2, 2\}$, namely the element $2$.
c. How many elements are in the set $\{0, \{0\}\}$ ?
There are two elements in the set, namely the symbol $0$, and the set $\{0\}$.
d. Is $\{0\} \in \{\{0\}, \{1\}\}$ ?
Yes, the set of $\{0\}$ is in the set $\{\{0\}, \{1\}\}$, as the set contains
both the sets $\{0\}$ and $\{1\}$.
e. Is $0 \in \{\{0\}, \{1\}\}$ ?
No, the symbol $0$, representing the number zero, is not in the set, which holds
two sets with the symbols in them.
5. Which of the following sets are equal?
$$
A = \{0, 1, 2\} \\
B = \{x \in \mathbb{R} | -1 \leq x < 3\} \\
C = \{x \in \mathbb{R} | -1 < x < 3\} \\
D = \{x \in \mathbb{Z} | -1 < x < 3\} \\
E = \{x \in \mathbb{Z}^+ | -1 < x < 3\}
$$
**Solution**
None of these sets are equal. $A = E$ might have worked had $A$ not included
$0$, but $E$ essentially evaluates to $E = \{1, 2\}$, and does not include $0$.
6. For each integer $n$, let $T_n = \{n, n^2\}$. How many elements are in each
of $T_2, T_{-3}, T_1$, and $T_0$? Justify your answers.
**Solution**
$$ T_2 = \{2, 2^2\} = \{2, 4\} \quad \text{ two elements } $$
$$ T_{-3} = \{-3, (-3)^2\} = \{-3, 9\} \quad \text{ two elements }$$
$$ T_1 = \{1, 1^2\} = \{1, 1\} = \{1\} \quad \text{ one element } $$
$$ T_0 = \{0, 0^2\} = \{0, 0\} = \{0\} \quad \text{ one element } $$
7. Use the set-roster notation to indicate the elements in each of the following
sets.
a. $S = \{n \in \mathbb{Z} | n = (-1)^k \text{, for some integer } k\}$
b. $T = \{m \in \mathbb{Z}| m = 1 + (-1)^i \text{, for some integer } i\}$
c. $U = \{r \in \mathbb{Z} | 2 \leq r \leq -2\}$
d. $V = \{s \in \mathbb{Z} | s > 2 \text{ or }x < 3\}$
e. $W = \{t \in \mathbb{Z} | 1 < t < -3\}$
f. $X = \{u \in \mathbb{Z} | u \leq 4 \text{ or } u \geq 1\}$
**Solution**
a. $S = \{n \in \mathbb{Z} | n = (-1)^k \text{, for some integer } k\}$
$$ \{-1, 1\} $$
b. $T = \{m \in \mathbb{Z}| m = 1 + (-1)^i \text{, for some integer } i\}$
$$ \{0, 2\} $$
c. $U = \{r \in \mathbb{Z} | 2 \leq r \leq -2\}$
$$ \emptyset $$
d. $V = \{s \in \mathbb{Z} | s > 2 \text{ or }x < 3\}$
$$ \mathbb{Z} $$
e. $W = \{t \in \mathbb{Z} | 1 < t < -3\}$
$$ \emptyset $$
f. $X = \{u \in \mathbb{Z} | u \leq 4 \text{ or } u \geq 1\}$
$$ \mathbb{Z}^+ $$
8. Let $A = \{c, d, f, g\}$, $B = \{f, j\}$, and $C = \{d, g\}$. Answer each of
the following questions. Give reasons for your answers.
a. Is $B \subseteq A$?
b. Is $C \subseteq A$?
c. Is $C \subseteq C$?
d. Is $C$ a proper subset of $A$?
**Solution**
a. Is $B \subseteq A$?
No, because every element of $B$ must be an element of $A$ by definition of a
subset, but $j \in B$, but $j \notin A$.
b. Is $C \subseteq A$?
Yes, every element of $C$ is an element of $A$.
c. Is $C \subseteq C$?
Yes, every element of $C$ is an element of $C$. By implication, every set is a
subset of itself.
d. Is $C$ a proper subset of $A$?
Yes, $C \subset A$, but $C \neq A$. Every element of $C$ is an element of $A$,
but $C$ does not equal $A$, which is the definition of a proper subset.
9.
a. Is $3 \in \{1, 2, 3\}$?
b. Is $1 \subseteq \{1}$?
c. Is $\{2\} \in \{1, 2\}$?
d. Is $\{3\} \in \{1, \{2\}, \{3\}\}$?
e. Is $1 \in \{1\}$?
f. Is $\{2\} \subseteq \{1, \{2\}, \{3\}\}$?
g. Is $\{1\} \subseteq \{1, 2\}$?
h. Is $1 \in \{\{1\}, 2\}$?
i. Is $\{1\} \subseteq \{1, \{2\}\}$?
j. Is $\{1\} \subseteq \{1\}$?
**Solution**
a. Is $3 \in \{1, 2, 3\}$?
Yes, the symbol $3$, representing the number three, is in the set $\{1, 2, 3\}$.
b. Is $1 \subseteq \{1}$?
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

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Page 25
A **universal statement** says that a certain property is true for all elements
in a set. (For example: _All positive numbers are greater than zero_.)
A **conditional statement** says that if one thing is true then some other thing
also has to be true. (For example: _If 378 is divisible by 18, then 378 is
divisible by 6_.)
Given a property that may or may not be true, an **existential statement** says
that there is at least one thing for which the property is true. (For example:
_There is a prime number that is even_.)
---
Page 30
**Set Roster Notation**
If $S$ is a set, the notation $x \in S$ means that $x$ is an element of $S$. The
notation $x \notin S$ means that $x$ is not an element of $S$. A set may be
specified using the **set-roster notation** by writing all of its elements
between braces. For example, $\{1, 2, 3\}$ denotes the set whose elements are
$1$, $2$, and $3$. A variation of the notation is sometimes used to describe a
very large set, as when we write $\{1, 2, 3, \dots, 100\}$ to refer to the set
of all integers from $1$ to $100$. A similar notation can also describe an
infinite set, as when we write $\{1, 2, 3, \dots\}$ to refer to the set of all
positive integers. (The symbol $\dots$ is called an **ellipsis** and is read
"and so forth.")
The **axiom of extension** says that a set is completely determined by what its
elements are - not the order in which they might be listed or the fact that some
elements might be listed more than once.
---
Page 30
| Symbol | Set |
| ---------- | --------------------------------------------------------- |
| \mathbb{R} | the set of all real numbers |
| \mathbb{Z} | the set of all integers |
| \mathbb{Q} | the set of all rational numbers, or quotients of integers |
---
Page 31
**Set-Builder Notation**
Let $S$ denote a set and let $P(x)$ be a property that elements of $S$ may or
may not satisfy. We may define a new set to be **the set of all elements $x$ in
$S$ such that $P(x)$ is true**. We denote this set as follows:
$$ \{x \in S | P(x)\} $$
Where $x$ is "the set of all" and $|$ is "such that."
---
Page 32
**Definition**
If $A$ and $B$ are sets, then $A$ is called a **subset** of $B$, written
$A \subseteq B$, if and only if, every element of $A$ is also an element of $B$.
Symbolically:
$A \subseteq B$ means that for every element $x$, if $x \in A$ then $x \in B$.
The phrases $A$ _is contained in_ $B$ and $B$ _contains_ $A$ are alternative
ways of saying that $A$ is a subset of $B$.
It follows from the definition of subset that for a set $A$ not to be a subset
of a set $B$ means that there is at least one element of $A$ that is not an
element of $B$. Symbolically:
$A \nsubseteq B$ means that there is at least one element $x$ such that
$x \in A$ and $x \notin B$.
**Definition**
Let $A$ and $B$ be sets. $A$ is a **proper subset** of $B$ if, and only if,
every element of $A$ is in $B$ but there is at least one element of $B$ that is
not in $A$.
---
Page 33
**Notation**
Given elements $a$ and $b$, the symbol $(a, b)$ denotes the **ordered pair**
consisting of $a$ and $b$ together with the specification that $a$ is the first
element of the pair and $b$ is the second element. Two ordered pairs $(a, b)$
and $(c, d)$ are equal if, and only if, $a = c$ and $b = d$. Symbolically:
$(a, b) = (c, d)$ means that $a = c$ and $b = d$.
---
Page 34
**Definition**
Let $n$ be a positive integer and let $x_1, x_2, \dots, x_n$ be (not necessarily
distinct) elements. The **ordered $n$-tuple, $(x_1, x_2, \dots, x_n)$**,
consists of $x_1$, $x_2$, $\dots$, $x_n$ together with the ordering: first
$x_1$, then $x_2$, and so forth up to $x_n$. An ordered 2-tuple is called an
**ordered pair**, and an ordered 3-tuple is called an **ordered triple.**
Two ordered $n$-tuples $(x_1, x_2, \dots, x_n)$ and $(y_1, y_2, \dots, y_n)$ are
**equal** if, and only if, $x_1 = y^1$, $x^2 = y^2$, $\dots$, and $x_n = y_n$.
Symbolically:
$$ (x_1, x_2, \dots x_n) = (y_1, y_2, \dots, y_n) \leftrightarrow x_1 = y_1, x^2 = y_2, \dots , x_n = y_n $$
---
Page 35
**Definition**
Given sets $A_1, A_2, \dots, A_n$, the **Cartesian product** of
$A_1, A_2, \dots, A_n$ denoted $A_1 \times A_2 \times \dots \times A_n$, is the
set of all ordered $n$-tuples $(a_1, a_2, \dots, a_n)$ where
$a_1 \in A_1, a_2 \in A_2, \dots , a_n \in A_n$.
Symbolically:
$$ A_1 \times A_2 \times \dots \times A_n = \{(a_1, a_2, \dots, a_n) | a_1 \in A_1, a_2 \in A_2, \dots , a_n \in A_n\} $$
In particular,
$$ A_1 \times A_2 = \{(a_1, a_2) | a_1 \in A_1 \text{ and } a_2 \in A_2\} $$
is the Cartesian product of $A_1$ and $A_2$.
---
Page 36
**Definition**
Let $n$ be a positive integer. Given a finite set $A$, a **string of length $n$
over $A$** is an ordered $n$-tuple of elements of $A$ written without
parentheses or commas. The elements of $A$ are called the **characters** of the
string. The **null string** over $A$ is defined to be the "string" with no
characters. It is often denoted $\lambda$ and is said to have length $0$. If
$A = \{0, 1\}$, then a string over $A$ is called a **bit string**.

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**Test Yourself**
Page 28
1. A universal statement asserts that a certain property is _______ for _______.
2. A conditional statement asserts that if one thing _______ then some other
thing _______.
3. Given a property that may or may not be true, an existential statement
asserts that _______ for which the property is true.
**Solutions**:
1. true, for all elements of a set.
2. is true, also has to be true.
3. there is at least one thing
---
**Test Yourself**
Page 37
1. When the elements of a set are given using the set-roster notation, the order
in which they are listed _______.
**Solution**
does not matter.
2. The symbol $\mathbb{R}$ denotes _______.
**Solution**
The set of all real numbers.
3. The symbol $\mathbb{Z}$ denotes _______.
**Solution**
The set of all integers.
4. The symbol $\mathbb{Q}$ denotes _______.
**Solution**
The set of all rational numbers.
5. The notation $\{x | P(x)\}$ is read _______.
**Solution**
The set of all $x$ such that $P(x)$ is true.
6. For a set $A$ to be a subset of a set $B$ means that _______.
**Solution**
Every element in $A$ is an element in $B$.
7. Given sets $A$ and $B$, the Cartesian product $A \times B$ is _______.
**Solution**
The set of all ordered pairs $(a, b)$ where $a \in A$ and $b \in B$.
8. Given sets $A$, $B$, and $C$, the Cartesian product $A \times B \times C$ is
_______.
The set of all ordered triples, $(a, b, c)$ where $a \in A$ and $b \in B$ and
$c \in C$.
**Solution**
9. A string of length $n$ over a set $S$ is an ordered $n$-tuple of elements
$S$, written without _______ or _______.
**Solution**
parentheses; commas

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