\begin{frame} \frametitle{One-To-One Functions} \begin{block}{} A \emph{one-to-one function} is a function that never takes the same value twice, that is: \begin{talign} f(x) \ne f(y) \quad \text{ whenever $x \ne y$} \end{talign} \end{block} \pause\medskip \begin{minipage}{.50\textwidth} \begin{center} \begin{tikzpicture}[default] \draw [fill=cblue!20,draw=cdblue] (0cm,0cm) to[out=10,in=90] (1cm,-1cm) to[out=-90,in=-10] (0cm,-2cm) to[out=170,in=-90,looseness=1.5] (-.2cm,-1cm) to[out=90,in=190,looseness=1.5] (0,0); \node (D) at (0,-2.6cm) {$D$}; \node (x) at (.2cm,-.4cm) {$a$}; \node (a) at (.4cm,-1cm) {$b$}; \node (z) at (.1cm,-1.7cm) {$z$}; \begin{scope}[xshift=35mm] \draw [fill=cblue!20,draw=cdblue,rotate=-20] (0cm,-1cm) ellipse (1cm and 1.3cm); \node (E) at (-.5,-2.6cm) {$E$}; \node (a') at (-.2cm,-.1cm) {$a$}; \node (c') at (-.4cm,-.8cm) {$b$}; \node (q') at (-.5cm,-1.7cm) {$d$}; \node (p') at (-0cm,-1.2cm) {$c$}; \end{scope} \begin{scope}[cdred,->,>=stealth,thick] \draw (x) to[bend left=20] (a'); \draw (a) to[bend left=-10] (a'); \draw (z) to[bend left=-20] (q'); \draw [shorten >= 5mm, shorten <= 5mm] (D) to node [above,black] {$f$} (E); \end{scope} \end{tikzpicture} \end{center} \end{minipage} \begin{minipage}{.49\textwidth} \begin{center} \begin{tikzpicture}[default] \draw [fill=cblue!20,draw=cdblue] (0cm,0cm) to[out=10,in=90] (1cm,-1cm) to[out=-90,in=-10] (0cm,-2cm) to[out=170,in=-90,looseness=1.5] (-.2cm,-1cm) to[out=90,in=190,looseness=1.5] (0,0); \node (D) at (0,-2.6cm) {$D$}; \node (x) at (.2cm,-.4cm) {$a$}; \node (a) at (.4cm,-1cm) {$b$}; \node (z) at (.1cm,-1.7cm) {$z$}; \begin{scope}[xshift=35mm] \draw [fill=cblue!20,draw=cdblue,rotate=-20] (0cm,-1cm) ellipse (1cm and 1.3cm); \node (E) at (-.5,-2.6cm) {$E$}; \node (a') at (-.2cm,-.1cm) {$a$}; \node (c') at (-.4cm,-.8cm) {$b$}; \node (q') at (-.5cm,-1.7cm) {$d$}; \node (p') at (-0cm,-1.2cm) {$c$}; \end{scope} \begin{scope}[cdred,->,>=stealth,thick] \draw (x) to[bend left=20] (a'); \draw (a) to[bend left=-10] (q'); \draw (z) to[bend left=-20] (c'); \draw [shorten >= 5mm, shorten <= 5mm] (D) to node [above,black] {$g$} (E); \end{scope} \end{tikzpicture} \end{center} \end{minipage} \pause\medskip \begin{exampleblock}{} Which of these function is one-to-one? \pause The function $g$. \end{exampleblock} \end{frame}