Exercises

Exercises 10.8 Exercises

1.

Is the set

\begin{equation*} \theta = \{((x, y), (5y, 4x, x + y)) : x, y \in\mathbb{R}\} \end{equation*}

a function? If so, what is its domain and its range?

2.

For which values of \(a,b\in\mathbb N\) does the set \(\phi=\set{(x,y)\in\mathbb Z\times \mathbb Z: ax+by= 6 }\) define a function?

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3.

Let \(f :\mathbb R\rightarrow \mathbb R\) be a function which is defined by \(f(x)=\dfrac{2x}{1+x^2}\text{.}\) Show that \(f(\mathbb R)=[-1,1]\text{.}\)

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4.

Consider the following functions and their images and preimages.

Consider the function \(f :\{1,2,3,4,5,6,7\}\rightarrow \{0,1,2,3,4,5,6,7,8,9\}\) given as

\begin{equation*} f =\{(1,3),(2,8),(3,3),(4,1),(5,2),(6,4),(7,6)\}. \end{equation*}

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Find: \(f\big(\{1,2,3\}\big)\text{,}\) \(f\big(\{4,5,6,7\}\big)\text{,}\) \(f(\emptyset)\text{,}\) \(f^{-1}\big(\{0,5,9\}\big)\) and \(f^{-1}\big(\{0,3,5,9\}\big)\text{.}\)
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Let \(g:\mathbb{R}\to \mathbb{R}\) be defined by \(g(x) = 4x^2-x-3\text{.}\)
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Find: \(g\left(\set{\frac{1}{8}}\right), g^{-1}(\set{0}), \, g((-1, 0)\cup [3, 4]),\, \text{ and, } g^{-1}([-10, -5))\text{.}\)
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Define \(h:\mathbb{R}\to \mathbb{R}\) by \(h(t) = \sin(2\pi t)\text{.}\)
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Find: \(h\left(\mathbb{Z}\right), \, h\left(\set{\frac{1}{4}, \frac{7}{2}, \frac{19}{4}, 22} \right),\, h^{-1}(\set{1}),\, \text{and } h^{-1}([0,1))\text{.}\)
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5.

Let \(A, B\) be sets and \(f:A\to B\) be a function from \(A\)to \(B\text{.}\) Then prove that if \(E\) and \(F\) are subsets of \(B\text{,}\) then

\begin{equation*} f^{-1}(E-F) = f^{-1}(E)-f^{-1}(F). \end{equation*}

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Remember that since we do not know whether or not \(f\) is a bijection, \(f^{-1}\) denotes the preimage of \(f\) not its inverse.

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6.

Let \(f:\mathbb R\to\mathbb R\) be the function defined by \(f(x) = x^2+ax+b\text{,}\) where \(a,b\in\mathbb R\text{.}\) Determine whether \(f\) is injective and/or surjective.

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7.

Determine all functions \(f:\mathbb N \rightarrow \mathbb N\) that are injective and such that for all \(n\in \mathbb N\) we have \(f(n)\leq n\text{.}\)

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8.

Prove that \(f:[3,\infty)\rightarrow [5,\infty)\text{,}\) defined by \(f(x)=x^2-6x+14\) is a bijective function.

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9.

For \(n\in\mathbb{N}\text{,}\) let \(A=\{ a_1, a_2, a_3,\cdots, a_n \}\) be a fixed set where \(a_j\neq a_i\) for any \(i\neq j\text{,}\) and let \(F\) be the set of all functions \(f: A\rightarrow \{ 0,1 \}\text{.}\)

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What is \(|F|\text{,}\) the cardinality of \(F\text{?}\)

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Now, for \(\mathcal{P}(A)\text{,}\) the power set of \(A\text{,}\) consider the function \(g:F \rightarrow \mathcal{P}(A)\text{,}\) defined as

\begin{equation*} g(f)=\{ a\in A: f(a)=1 \}. \end{equation*}

Is \(g\) injective? Is \(g\) surjective?

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10.

Let \(f : \mathbb{Z} \rightarrow \mathbb{Z}\times\mathbb{Z}\) be defined by \(f(n) = (2n+1, n + 2)\text{.}\) Check whether this function is injective and whether it is surjective. Prove your answer.

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11.

Let \(f:E\rightarrow F\) be a function. We recall that for any \(A\subseteq E\) the image of \(A\) by \(f\text{,}\) namely \(f(A)\text{,}\) is defined as

\begin{gather*} f(A)=\{f(x): x\in A\}. \end{gather*}

Show that \(f\) is surjective if and only if

\begin{gather*} \forall A\in \mathcal P(E),\ F-f(A)\subseteq f(E-A). \end{gather*}

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12.

Let \(f:C\to D\) be a function. Let \(A, B \subset C\) be nonempty sets. Prove that if \(f\) is injective, then \(f(A-B)=f(A)-f(B)\text{.}\)

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13.

Consider the function

\begin{equation*} f: \mathbb N\times\mathbb N\rightarrow \mathbb N \qquad \text{where} \qquad f(x,y) = 2^{x-1}(2y-1). \end{equation*}

Prove that \(f\) is bijective.

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14.

Let \(A, B\) be nonempty sets. Prove that if there is a bijection \(f:A\rightarrow B\text{,}\) then there is a bijection from \(\mathcal{P}(A)\text{,}\) the power set of \(A\text{,}\) to \(\mathcal{P}(B)\text{,}\) the power set of \(B\text{.}\)

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15.

Let \(\rel\) be the relation on \(\mathbb R^2\) defined by

\begin{equation*} (x,y)\rel (s,t) \text{ iff } x^2+y^2=s^2+t^2 \end{equation*}

where \((x,y), (s,t) \in \mathbb{R}^2\text{.}\)

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Show that \(\rel\) is an equivalence relation
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Let \(\mathcal S\) be the set of equivalence classes of the relation \(\rel\) defined in part (a). Let \(f\) be defined by

\begin{equation*} f:\mathcal S\to [0,\infty) \qquad \text{ with } \qquad f([(x,y)])=\sqrt{x^2+y^2}. \end{equation*}

Prove that
- \(f\) is a function, and, further
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- \(f\) is bijective.
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16.

Let \(n\in\mathbb{N}\) with \(n\gt 1\) and let \(\mathbb{Z}_n\) be the set of equivalence classes modulo \(n\text{.}\) For any \(x\in\mathbb{Z}\text{,}\) let \([x]_n \in \mathbb{Z}_n\) denote its equivalence class modulo \(n\text{.}\)

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Define the function \(f:\mathbb{Z}_n\to \set{0,1,\dots,n-1}\) by \(f([x]_n)=r\text{,}\) where \(r\) is the remainder of \(x\) upon division by \(n\text{.}\)

Show that \(f\) is well-defined, meaning that \(f\) is defined on its whole domain and that \(f\) does not depend on the choice of representative for each equivalence class; i.e. \([x]_n=[y]_n \implies f([x]_n)=f([y]_n)\text{.}\)
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Show that \(f\) is a bijection.
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This question explains why when dealing with equivalence classes of integers modulo \(n\text{,}\) we often consider the set of representatives \(\set{0,1,\dots,n-1}\) instead.

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17.

Consider \(f : A\rightarrow B\text{.}\) Prove that f is injective if and only if \(X = f^{-1}( f (X))\) for all \(X \subseteq A\text{.}\)

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18.

Consider \(f : A\rightarrow B\text{.}\) Prove that f is surjective if and only if \(f ( f^{-1}(Y)) = Y\) for all \(Y \subseteq B\text{.}\)

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19.

We say that a function \(f:\mathbb{R}\to \mathbb{R}\) is strictly increasing if whenever \(x_1 \lt x_2\) we have \(f(x_1) \lt f(x_2)\text{.}\) Similarly, a function \(g: \mathbb{R}\to \mathbb{R}\) is strictly decreasing if whenever \(x_1 \lt x_2\) we have \(g(x_1) \gt g(x_2)\text{.}\)

Prove that the composition of two strictly increasing functions is strictly increasing.
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Prove that the composition of two strictly decreasing functions is strictly increasing.
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20.

Let \(f,g\text{,}\) and \(h\) be functions from \(\mathbb{R}\to \mathbb{R}\text{.}\) We define the following operations on functions:

Function addition: \(\left(f+g\right)(x) = f(x)+g(x)\)
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Function division: \(\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}\)
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Function composition: \(\left(f\circ g\right)(x) = f(g(x))\)
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Note that under this definition, \(\left(\frac{1}{f}\right)(x) = \frac{1}{f(x)}\text{.}\) This is NOT the inverse of \(f(x)\text{.}\)

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Prove or give a counterexample to the following statements:

\(f\circ (g+h)= f\circ g + f\circ h\) for every \(x\in \mathbb{R}\text{.}\)
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\((g+h)\circ f = g\circ f + h\circ f\) for every \(x\in \mathbb{R}\text{.}\)
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\(\frac{1}{f\circ g} = \frac{1}{f}\circ g\) for every \(x\in \mathbb{R}\text{.}\)
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\(\frac{1}{f\circ g} = f \circ \frac{1}{g}\) for every \(x\in \mathbb{R}\text{.}\)
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21.

Find counterexamples to the following statements:

Given a function \(f : A\rightarrow B\) and subsets \(W,X \subseteq A\text{,}\) we have \(f (W\cap X) = f (W)\cap f (X)\text{.}\)
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Given a function \(f : A \rightarrow B\) and a subset \(Y \subseteq B\text{,}\) we have \(f ( f^{-1}(Y)) = Y\text{.}\)
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Explain your answers.

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22.

Let \(A, B\) be nonempty sets. Let \(f, h\) be functions from \(A\) to \(B\text{,}\) and let \(g\) be a function from \(B\) to \(A\text{.}\)

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Suppose that \(g\) is injective. Prove that if \(g\circ f=g\circ h\text{,}\) then \(f=h\text{.}\)
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Suppose that \(g\) is surjective. Prove that if \(f \circ g=h\circ g\text{,}\) then \(f=h\text{.}\)
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23.

Consider the following functions and their compositions.

Let \(f:\mathbb{R}\to \mathbb{R}\) be defined by \(f(x)=x+1\text{.}\) Does there exist a function \(g:\mathbb{R}\to \mathbb{R}\) such that \((f\circ g )(x) = (g\circ f)(x)\) for every \(x\in \mathbb{R}\text{?}\)
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Let \(f:\mathbb{R}\to \mathbb{R}\) be defined by \(f(x)=c\) for some \(c\in \mathbb{R}\) (i.e. \(f\) is a constant function). Which functions \(g:\mathbb{R}\to \mathbb{R}\) have the property \((f\circ g) (x) = (g\circ f)(x)\) for every \(x\in \mathbb{R}\text{?}\)
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Let \(f:\mathbb{R}\to \mathbb{R}\text{.}\) Suppose \((f\circ g)(x) = (g\circ f)(x)\) for every \(x\in \mathbb{R}\) and for every function \(g:\mathbb{R}\to \mathbb{R}\text{.}\) Show that \(f(x) = x\text{.}\)
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24.

Let \(A\) be a nonempty set and \(f:A\rightarrow A\) be a function.

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Prove that \(f\) is bijective if \(f\circ f\) is bijective.
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Let

\begin{equation*} f:(0,\infty) \to (0,\infty) \qquad f(x) = \log\left( \dfrac{e^x+1}{e^x-1} \right) \end{equation*}

where \(\log(x) \) denotes the natural logarithm of \(x\text{.}\) Use part (a) to prove that this is a bijective function.

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25.

Prove that the function

\begin{equation*} f : \mathbb R-\set{-2}\rightarrow\mathbb R-\set{1}, \text{ defined by } f(x)=\dfrac{x+1}{x+2} \end{equation*}

is bijective and find its inverse.

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26.

Let \(f:\mathbb Z \rightarrow \mathbb Z\) defined so that \(f(n)=-n+3\) if \(n\) is even and \(f(n)=n+7\) if \(n\) is odd. Prove that \(f\) is bijective and find its inverse, \(f^{-1}\text{.}\)

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27.

The following question concerns the triple composition of a function.

Let \(A\) be a non-empty set and let \(g:A \to A\) be a function that satisfies \(g\circ g\circ g=i_A\text{,}\) where \(i_A\) is the identity function on the set \(A\text{.}\) Prove that \(g\) must be bijective.
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Let \(A=\mathbb R -\{0,1\}\) and let \(f:A\to A\) be defined by \(f(x)=1-\frac{1}{x}\text{.}\) Show that \(f\circ f\circ f = i_A\) .
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Use part (a) to conclude that \(f\) is bijective and determine its inverse function \(f^{-1}\text{.}\)
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