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\flushleft\underline{Homework 9 --- MATH 2510 Spring 2018} \\
Recall that the notation $f \colon A \rightarrow B$ is how you say that there is a function named ``$f$" whose domain (i.e. set of inputs) is ``$A$" and whose codomain (i.e. space where outputs live) is ``$B$". Also recall that the range of a function is the set of actual outputs, i.e.
$$\mathrm{range}(f) = \{y \in B \colon \exists x f(x)=y\}.$$
We defined a function $f \colon X \rightarrow Y$ to be one-to-one provided that $\forall x \forall y (f(x)=f(y) \rightarrow x=y)$. We defined the cardinality (i.e. ``number of elements") $|A|$ of a set $A$ in the following way: we say that $|X| \leq |Y|$ provided there is a one-to-one function $f \colon X \rightarrow Y$. We say that $|X|=|Y|$ (i.e. ``the cardinality of $X$ is equal to the cardinality of $Y$") provided that $|X| \leq |Y|$ and $|Y| \leq |X|$. Of course if a set is finite, we simply report its size as an number; e.g. $|\{a,b,c\}|=3$.
\begin{enumerate}[1.]
\item Sketch each function (to the best of your abilities). Is the function one-to-one or not?
\begin{enumerate}[a.)]
\item $\left\{\begin{array}{ll}
f \colon \mathbb{N} \rightarrow \mathbb{N} \\
f(n)=n+1
\end{array} \right.$
\item $\left\{\begin{array}{ll}
f \colon \mathbb{Z} \rightarrow \mathbb{Z} \\
f(n)=n+1
\end{array} \right.$
\item $\left\{\begin{array}{ll}
f \colon \mathbb{Q} \rightarrow \mathbb{Q} \\
f(n)=\dfrac{1}{x^2+1}
\end{array} \right.$
\item $\left\{\begin{array}{ll}
f \colon \mathbb{R}\setminus \{0\} \rightarrow \mathbb{R} \\
f(x)=\dfrac{1}{x^2}
\end{array} \right.$
\item $\left\{\begin{array}{ll}
f \colon [0,\infty) \rightarrow \mathbb{R} \\
f(x)=\dfrac{1}{1+|x|}
\end{array} \right.$
\end{enumerate}
\item Show that $|\mathbb{Z}| = |\mathbb{N}|$. 
\item Show that $|\mathbb{N} \times \mathbb{N} \times \mathbb{N}|=|\mathbb{N}|$. (Recall how the ``$\times$" symbol works:
$$A \times B \times C = \{ (a,b,c) \colon a \in A, b \in B, c \in C \}.	)$$
\item Show that $|(0,1]|=|[1,\infty)|$ (note: these are intervals on the real line). (\textit{hint: consider \href{https://en.wikipedia.org/wiki/Rational_function}{rational functions}})
\item Show that $\left| \left( - \dfrac{\pi}{2}, \dfrac{\pi}{2} \right) \right| = |\mathbb{R}|$. (\textit{hint: consider \href{https://en.wikipedia.org/wiki/Inverse_trigonometric_functions}{inverse trigonometric functions}})
\item Find the cardinality of the set of functions $f \colon \{a,b\} \rightarrow \{c,d\}$ (i.e. how many such functions are there? brute force of listing all of them can get you the answer!). 

\end{enumerate}
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