\documentclass[]{webquiz} \BreadCrumbs{quizindex | title} \DeclareMathOperator{\cis}{cis} \newcommand{\R}{\mathbb R} \newcommand{\C}{\mathbb C} \usepackage{pst-all} \title{Quiz 1: Numbers and sets} \begin{document} \begin{question} Which of the following are correct ways of writing the set \[ A=\{x \in \mathbb{R} \mid -3 < x \leq -1 \text{ or } x\geq 0\} ?\] \begin{choice}[columns=2] \incorrect $(-3,\infty)$ \feedback The interval $(-3,\infty)$ includes the real numbers between $-1$ and $0$, which do not belong to $A$. \incorrect $[-3,\infty)$ \feedback The interval $[-3,\infty)$ includes $-3$, and the real numbers between $-1$ and $0$, which do not belong to $A$. \incorrect $[-3,-1]\cap[0,\infty)$ \feedback The interval $[-3,-1]\cap[0,\infty)$ is the empty set $\emptyset$. As $A$ is not empty (for example, $A$ includes $-1$), this option cannot be correct. \incorrect $(-3,-1)\cup[0,\infty)$ \feedback The interval $-1$ is not in $(-3,-1)\cup[0,\infty)$, but $-1$ is in $A$. \correct $(-3,-1]\cup[0,\infty)$ \end{choice} \end{question} %%%% 2 \begin{question} What is another way of writing the set \[B= \{x \in {\mathbb R}\ |\ |x-3|<2 \}\ \rm{?}\] \begin{choice}[columns=2] \incorrect $(2,3]$ \feedback For example, $4$ belongs to $B$ but is not in $(2,3]$. \incorrect $[2,4]$ \feedback For example, $1.5$ belongs to $B$ but is not in $[2,4]$. \correct $(1,5)$ \feedback $B$ is the set of all points whose distance from 3 on the number line is less than 2. \\ The solution to $|x-3|<2$ is $1}(0,0)(-1.5,-1.5)(3.5,3.5) \rput(3.75,0){$x$} \rput(0,3.85){$iy$} \rput(3,-0.4){3} \rput(-0.4,3){3$i$} \psdots(1,1) \end{pspicture} \end{center} corresponds to which set of complex numbers? \begin{choice}[columns=2] \correct\(\{z \in \C : |z-(i+1)|<2\}\) \incorrect \(\{z \in \C : |z|-|1+i|<2\}\) \feedback This set corresponds to the interior of a circle, centre the origin, radius $2+\sqrt 2$. \incorrect \(\{z \in \C : \text{Re}(z+(i+1))<2 \}\) \feedback This set corresponds to the open half plane containing all complex numbers $z=x+iy$ with $x<1$. \incorrect \(\{z \in \C : |z-2|<|i+1-2|\}\) \feedback This set corresponds to the interior of a circle, centre $2$, radius $\sqrt 2$. \incorrect None of the above. \end{choice} \end{question} \end{document}