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\flushleft \underline{Homework 8 -- MATH 2510 Spring 2019}\\
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An ordered pair $(a,b)$ is the set $\{a,\{a,b\}\}$. A relation is a set of ordered pairs. A function $f$ is a relation with the property that if $(a,b) \in f$ and $(a,c) \in f$, then $b=c$. We write $f \colon A \rightarrow B$ to mean that $f$ is a function with domain (``where inputs live") $A$ and (``where outputs live") codomain $B$. The range of a function is  given by $\mathrm{range}(f)=\{y \colon \exists x (x,y) \in f\}$. A one-to-one function is a function $f$ with the property that if $(a,b) \in f$ and $(c,b) \in f$, then $a=c$ (``no output has two different inputs").  We say that $\mathrm{card}(X) \preceq \mathrm{card}(Y)$ if there is a one-to-one function $f \colon X \rightarrow Y$. We say that $\mathrm{card}(X)=\mathrm{card}(Y)$ whenever both $X \preceq Y$ and $Y \preceq X$. 

\begin{enumerate}[1.]
\item Is the function $\left\{ \begin{array}{ll}
f \colon \mathbb{N} \rightarrow \mathbb{N} \\
f(n)=2n
\end{array} \right.$ one-to-one? Draw/sketch it (software ok) and explain why or why not.
\item Is the function $\left\{ \begin{array}{ll}
f \colon [-1,1] \rightarrow \mathbb{R} \\
f(x) = (x+1)^2
\end{array} \right.$ one-to-one? Draw/sketch it (software ok) and explain why or why not.
\item Is the function $\left\{ \begin{array}{ll}
f \colon (0,\infty) \rightarrow \mathbb{R} \\
f(x)=x+\ln(x)
\end{array} \right.$ one-to-one? Draw/sketch it (software ok) and explain why or why not.
\item Is the function $\left\{ \begin{array}{ll}
f \colon \left(\dfrac{\pi}{2}, \pi \right) \rightarrow \mathbb{R} \\
f(x)=\sin \left( \dfrac{x^2+1}{x^3} \right)
\end{array} \right.$ one-to-one? Draw/sketch it (software ok) and explain why or why not.
\item Consider the set $\mathbb{P}_2$ of polynomials of degree $\leq 2$ and $\mathbb{P}_1$, i.e. $\mathbb{P}_2=\{ax^2+bx+c \colon a,b,c \in \mathbb{R}\}$. Define the function \[\left\{ \begin{array}{ll}
D \colon \mathbb{P}_2 \rightarrow \mathbb{P}_2 \\
\begin{array}{ll}
D(ax+b) &= \left( \dfrac{\mathrm{d^2}}{\mathrm{d}x^2}+x \right)(ax^2+bx+c) \\
&= (ax^2+bx+c)''+x(ax^2+bx+c).
\end{array} 
\end{array} \right.\]
Is this function one-to-one? Why or why not?
\item Define a function (using symbols or a picture) to show that $\mathrm{card}\left( \{0,2,4,6,8,10,\ldots\} \right) \preceq \mathrm{card}\left(\{0,1,2,3,4,5,\ldots\}\right)$.
\item Define a function (using symbols or a picture) to show that $\mathrm{card}(\mathbb{Z}) \preceq \mathrm{card}(\mathbb{R})$. 
\item Show that $\mathrm{card}\Big( (-2,3) \Big) \preceq \mathrm{card} (\mathbb{R})$. (hint: \textit{think about ``rational functions"}). Write \textbf{formulas} for your functions and draw.
\item Show that $\mathrm{card} \left( \dfrac{-\pi}{2}, \dfrac{\pi}{2} \right) = \mathrm{card}(\mathbb{R})$ (hint: \textit{inverse trig functions}). Write \textbf{formulas} for your functions and draw.
\end{enumerate}
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