complete the questions in latex format

% CSC165 Winter 2023: Problem Set 0 % Template to use to complete Problem Set 0. % Note that using this template is *optional*: it provides a nice foundation % for getting started with LaTeX, but you aren't required to use it! % If you are using Overleaf, you'll want to upload this file to your account. % Before modifying this file, we recommend trying to compile it as-is % to see what the basic template gives.

\documentclass[12pt]{article}

\usepackage{amsmath} \usepackage[margin=2.5cm]{geometry}

% If you want to use this package, make sure to download it from the LaTeX % help page on Quercus (and copy to Overleaf, if using)! Then remove the % comment indicator % that starts the next line. % \usepackage{csc}

% Document metadata \title{Insert title here} \author{Insert author here} \date{Insert date here}

% Document starts here \begin{document} \maketitle

\section*{My Courses}

% Use the environment "itemize" to create an *unordered* list.

Remove this line and replace it with a list of the courses you're taking.

\section*{Set notation}

% Fill in the following:

\[ S_1 \cap S_2 = \dots \]

\section*{A truth table}

Remove this line after following this Hint: Look at {\texttt sample\_latex.tex} for an example of a table using {\texttt tabular}.

\section*{A calculation}

Remove this line after following this Hint: Look at {\texttt sample\_latex.tex} for an example of {\texttt align*}.

\begin{align*} \sum_{i=0}^{n-1} (2i + 3) &= \dots\\ % TODO: fill in steps here! \end{align*}

\end{document}

ABOVE IS THE GIVEN FORMATTING

1. Create a title for your document that shows the course code (CSC165H1), the phrase "Problem Set 0", your name, and a date (this can either be the problem set due date or the LaTeX command \today). 2. Use the itemize environment to create an unordered list showing the courses you are taking this term. Each list item should contain the course's code, title, and the name of your instructor. 3. Definition. Let nN. We say that n is a magical number if it can be written in the form n=5i+j for some iN and j{1,0,1}. Let S1 be the set of all magical numbers, and S2 be the set of all integers less than 15. List all of the elements in the intersection of S1 and S2 using set notation (i.e., S1S2={}, where you fill in the dots). Note that 0 is a natural number. 4. Write down the truth table for the propositional formula (pq)(pr). There should be eight rows in your table (plus a header row). Make sure to use the tabular environment to create a table in LaTeX. You may, but are not required to, show "intermediate columns" (e.g., for pq ) in your truth table. 5. Finally, show the steps involved to solve the following problem: find the smallest positive integer n such that i=0n1(2i+3) is greater than or equal to 165 (without "guessing and checking" or writing a computer program!). Your solution should consist of two parts: (i) Use the align* environment to simplify this summation (show at least two or three steps). You may use the following arithmetic series formula, which is valid for all nN and all d,kR : i=0n1(di+k)=nk+2dn(n1) (ii) Use the align* environment to manipulate an inequality. You will probably find the quadratic formula helpful. In both of the above parts, you should make use of the \& symbol in the align* environment to vertically align the =,, and/or symbols in your calculations