🚧 Almost done with chapter 1
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@ -13,7 +13,7 @@ b. Given any real number, its square is nonnegative.
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a. Are there numbers $a$ and $b$ with the property that $a^2 + b^2 = (a + b)^2$?
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_Or_: Are there numbers $a$$ and $b$ such that $a^2 + b^2 = (a + b)^2$?
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_Or_: Are there numbers $a$ and $b$ such that $a^2 + b^2 = (a + b)^2$?
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_Or_: Do there exist any numbers $a$ and $b$ such that $a^2 + b^2 = (a + b)^2$?
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@ -422,3 +422,296 @@ data structures and to respect the notations that are used for them. Similarly
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in mathematics, it is important to distinguish among, say, _{a, b, c}, {{ab},
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c}, (a, b, c), (a, (b, c)), abc_ and so forth, because these are all
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significantly different objects.
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---
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**Example 1.3.1 A Relation as a Subset**
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Page 39
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Let $A = \{1, 2\}$ and $B = \{1, 2, 3\}$ and define a relation $R$ from $A$ to
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$B$ as follows: Given any $(x, y) \in A \times B$.
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$(x, y) \in R$ means that $\dfrac{x - y}{2}$ is an integer.
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a. State explicitly which ordered pairs are in $A \times B$ and which are in
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$R$.
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b. Is 1 _R_ 3? Is 2 _R_ 3? Is 2 _R_ 2?
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c. What are the domain and co-domain of _R_?
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**Solution**
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a. State explicitly which ordered pairs are in $A \times B$ and which are in
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$R$.
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$$ A \times B = \{(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3)\} $$
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$$ (x, y) \in R = \{(A \times B) | \left(\frac{x - y}{2}\right) \in \mathbb{Z}\} $$
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$$ R = \{(1, 1), (1, 3), (2, 2)\} $$
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b. Is 1 _R_ 3? Is 2 _R_ 3? Is 2 _R_ 2?
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Is 1 _R_ 3?: Yes, because $(1, 3) \in R$.
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Is 2 _R_ 3? No, because $(2, 3) \notin R$.
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Is 2 _R_ 2? Yes, because $(2, 2) \in R$.
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c. What are the domain and co-domain of _R_?
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The domain of _R_ is $\{1, 2\}$ and the co-domain of _R_ is $\{1, 2, 3\}$
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---
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**Example 1.3.2 The Circle Relation**
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Page 40
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Define a relation $C$ from $\mathbb{R}$ to $\mathbb{R}$ as follows: For any
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$(x, y) \in \mathbb{R} \times \mathbb{R}$.
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$(x, y) \in C$ means that $x^2 + y^2 = 1$.
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a. Is $(1, 0) \in C$? Is $(0, 0) \in C$? Is
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$\left(-\dfrac{1}{2}, \dfrac{\sqrt{3}}{2}\right) \in C$? Is -2 _C_ 0? Is 0 _C_
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(-1)? Is 1 _C_ 1?
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b. What are the domain and co-domain of _C_?
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c. Draw a graph for _C_ by plotting the points of _C_ in the Cartesian plane.
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**Solution**
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a. Is $(1, 0) \in C$? Is $(0, 0) \in C$? Is
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$\left(-\dfrac{1}{2}, \dfrac{\sqrt{3}}{2}\right) \in C$? Is -2 _C_ 0? Is 0 _C_
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(-1)? Is 1 _C_ 1?
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Is $(1, 0) \in C$?
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Yes, $(1)^2 + (0)^2 = 1$
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Is $(0, 0) \in C$?
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No, $(0)^2 + (0)^2 = 0 \neq 1$
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Is $\left(-\dfrac{1}{2}, \dfrac{\sqrt{3}}{2}\right) \in C$?
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Yes,
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$\left(-\frac{1}{2}\right)^2 + \left(\frac{\sqrt{3}}{2}\right)^2 = \frac{1}{4} + \frac{3}{4} = 1$
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Is -2 _C_ 0?
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No because $(-2)^2 + (0)^2 = 4 \neq 1$
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Is 0 _C_ (-1)?
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Yes because $(0)^2 + (-1)^2 = 1$.
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Is 1 _C_ 1?
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No, because $(1)^2 + (1)^2 = 2 \neq 1$.
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b. What are the domain and co-domain of _C_?
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The domain of _C_ is $\mathbb{R}$ and the co-domain of _C_ is also $\mathbb{R}$.
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c. Draw a graph for _C_ by plotting the points of _C_ in the Cartesian plane.
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This is just the circle formula, so:
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---
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**Example 1.3.3 Arrow Diagrams and Relations**
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Page 41
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Let $A = \{1, 2, 3\}$ and $B = \{1, 2, 3\}$ and define relations $S$ and $T$
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from $A$ to $B$ as follows:
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For every $(x, y) \in A \times B$,
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$(x, y) \in S$ means that $x < y$ ($S$ is a "less than" relation).
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$T = \{(2, 1), (2, 5)\}$.
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Draw arrow diagrams for $S$ and $T$.
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**Solution**
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These example relations illustrate that it is possible to have several arrows
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coming out of the same element of $A$ pointing in different directions. Also, it
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is quite possible to have an element of $A$ that does not have an arrow coming
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out of it.
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---
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**Example 1.3.4 Functions and Relations on Finite Sets**
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Page 42
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Let $A = \{2, 4, 6\}$ and $B = \{1, 3, 5\}$. Which of the relations $R$, $S$,
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and $T$ defined below are functions from $A$ to $B$?
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a. $R = \{(2, 5), (4, 1), (4, 3), (6, 5)\}$
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b. For every $(x, y) \in A \times B$, $(x, y) \in S$ means that $y = x + 1$.
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c. $T$ is defined by the arrow diagram
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**Solution**
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a. $R = \{(2, 5), (4, 1), (4, 3), (6, 5)\}$
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$R$ is not a function because it does not satisfy property (2). The ordered
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pairs $(4, 1)$ and $(4, 3)$ have the same first element but different second
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elements. You can see this graphically if you draw the arrow diagram for $R$.
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There are two arrows coming out of 4: One point to 1 and the other points to 3.
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b. For every $(x, y) \in A \times B$, $(x, y) \in S$ means that $y = x + 1$.
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$S$ is not a function because it does not satisfy property (1). It is not true
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that every element of $A$ is the first element of an ordered pair in $S$. For
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example $6 \in A$ but there is no $y$ in $B$ such that $y = 6 + 1 = 7$. You can
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also see this graphically by drawing the arrow diagram for $S$.
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c. $T$ is defined by the arrow diagram
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$T$ is a function: Each element in $\{2, 4, 6\}$ is related to some element in
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$\{1, 3, 5\}$, and no element in $\{2, 4, 6\}$ is related to more than one
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element in $\{1, 3, 5\}$. When these properties are stated in terms of the arrow
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diagram, they become (1) there is an arrow coming out of each element of the
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domain, and (2) no element of the domain has more than one arrow coming out of
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it. So you can write $T(2) = 5$, $T(4) = 1$, $T(6) = 1$.
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---
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**Example 1.3.5 Functions and Relations on Sets of Strings**
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Page 43
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Let $A = \{a, b\}$ and let $S$ be the set of all strings over $A$.
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a. Define a relation $L$ from $S$ to $\mathbb{Z}^{\text{nonneg}}$ as follows:
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For every string $s$ in $S$ and for every nonnegative integer $n$,
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$$ (s, n) \in L \text{ means that the length of } s \text{ is } n $$
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Observe that $L$ is a function because every string in $S$ has one and only one
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length. Find _L(abaaba)_ and _L(bbb)_.
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b. Define a relation $C$ from $S$ to $S$ as follows: For all strings $s$ and $t$
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in $S$,
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$$ (s, t) \in C \text{ means that } t = as $$
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where $as$ is the string obtained by appending $a$ on the left of the characters
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in $s$. ($C$ is called **concatenation** by $a$ on the left.) Observe that $C$
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is a function because every string in $S$ consists entirely of $a$'s and $b$'s
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and adding an additional $a$ on the left creates a new string that also consists
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of $a$'s and $b$'s and thus is also in $S$. Find _C(abaaba)_ and _C(bbb)_.
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**Solution**
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a. Define a relation $L$ from $S$ to $\mathbb{Z}^{\text{nonneg}}$ as follows:
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For every string $s$ in $S$ and for every nonnegative integer $n$,
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$$ (s, n) \in L \text{ means that the length of } s \text{ is } n $$
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Observe that $L$ is a function because every string in $S$ has one and only one
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length. Find _L(abaaba)_ and _L(bbb)_.
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_L(abaaba)_ = 6
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_L(bbb)_ = 3
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b. Define a relation $C$ from $S$ to $S$ as follows: For all strings $s$ and $t$
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in $S$,
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$$ (s, t) \in C \text{ means that } t = as $$
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where $as$ is the string obtained by appending $a$ on the left of the characters
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in $s$. ($C$ is called **concatenation** by $a$ on the left.) Observe that $C$
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is a function because every string in $S$ consists entirely of $a$'s and $b$'s
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and adding an additional $a$ on the left creates a new string that also consists
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of $a$'s and $b$'s and thus is also in $S$. Find _C(abaaba)_ and _C(bbb)_.
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_C(abaaba)_ = aabaaba
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_C(bbb)_ = abbb
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---
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**Example 1.3.6 Functions Defined by Formulas**
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Page 44
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The **squaring function** $f$ from $\mathbb{R}$ to $\mathbb{R}$ is defined by
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the formula $f(x) = x^2$ for every real number $x$. This means that no matter
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what real number input is substituted for $x$, the output of $f$ will be the
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square of that number. The idea can be represented by writing
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$f(\Box) = \Box^2$. In other words, $f$ sends each real number $x$ to $x^2$, or
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symbolically, $f: x \to x^2$. Note that the variable $x$ is a dummy variable;
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any other symbol could replace it, as long as the replacement is made everywhere
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the $x$ appears.
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The **successor function** $g$ from $\mathbb{Z}$ to $\mathbb{Z}$ is defined by
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the formula $g(n) = n + 1$. Thus, no matter what integer is substituted for $n$,
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the output of $g$ will be that number plus $1$: $g(\Box) = \Box + 1$. In other
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words, $g$ sends each integer $n$ to $n + 1$, or, symbolically,
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$g: n \to n + 1$.
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An example of a **constant function** is the function $h$ from $\mathbb{Q}$ to
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$\mathbb{Z}$ defined by the formula $h(r) = 2$ for all rational numbers $r$.
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This function sends each rational number $r$ to $2$. In other words, no matter
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what the input, the output is always $2$: $h(\Box) = 2$ or $h: r \to 2$.
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The functions $f$, $g$, and $h$, are represented by the function machines in
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Figure 1.3.2 (see page 44).
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A function is an entity in its own right. It can be thought of as a certain
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relationship between sets or as an input/output machine that operates according
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to a certain rule. This is the reason why a function is generally denoted by a
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single symbol or string of symbols, such as $f$, $G$, or $\log$, or $\sin$.
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A relation is a subset of a Cartesian product and a function is a special kind
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of relation. Specifically, if $f$ and $g$ are functions from a set $A$ to a set
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$B$, then
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$$ f = \{(x, y) \in A \times B | y = f(x)\} \quad \text{ and } g(x) = \{(x, y) \in A \times B | y = g(x)\} $$
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It follows that
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$$ f \text{ equals } g, \quad \text{ written } f = g, \quad \text{ if, and only if, } f(x) = g(x) \text{ for all } x \text{ in } A $$
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---
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**Example 1.3.7 Equality of Functions**
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Page 44
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Define functions $f$ and $g$ from $\mathbb{R}$ to $\mathbb{R}$ by the following
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formulas:
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$$ f(x) = |x| \quad \text{ for every } x \in \mathbb{R} $$
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$$ g(x) = \sqrt{x^2} \quad \text{ for every } x \in \mathbb{R} $$
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Does $f = g$?
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**Solution**
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Yes. Because the absolute value of any real number equals the square root of its
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square, $|x| = \sqrt{x^2}$ for all $x \in \mathbb{R}$. Hence $f = g$.
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