Btukfyl

It was hard for me to come up with a title for this question. So, for one, my document is starting on the second page. The second issue is that I cannot seem to get everything to appear in my document. I have ten questions, but only 8 and part of the 9th show up. When I remove questions, the 10th one partially shows up. It's as if the entire document is trying to force itself all onto one page. Any help would be greatly appreciated. The code is below. I'm stumped. Thank you so much.

documentclass[a4paper, 12pt]{article}

usepackage{amsmath,amsthm,amssymb,amsfonts, enumitem, fancyhdr, color, comment, graphicx, environ}

newcommand{powerset}[1]{mathbb{P}(#1)}

setcounter{secnumdepth}{-2}

usepackage{parskip}

usepackage{tikz}

setlength{parindent}{15pt}



begin{document}



section{fbox{1} textnormal{Let emph{A} = {1, 2, 3}. How many relations are there on A?}}





section{fbox{2} textnormal{Let emph{R} be a relation from sets emph{A} to emph{B}. Prove that Rng($R^{-1}$) = Dom(R).}}





section{fbox{3} textnormal{Let emph{A} = {a, b, 1, 2, 3}. Define emph{R} = {(1, 1), (a, a), (a, b), (1, 2), (2, 1), (a, 1), (a, 2)}. Is this relation reflexive, symmetric, transitive, irreflexive, and/or antisymmetric? Explain. Draw a digraph of emph{R}.}}





section{fbox{4} textnormal{Let emph{A} = {a, b, 1, 2, 3}. Let emph{T} be an equivalence relation on emph{A} such that aT2 and bT3. How many equivalence classes of emph{T} are possible? List the equivalence classes for all possibilities.}}





section{fbox{5} textnormal{Let emph{Q} = {(x, y) $in mathbb{R} times mathbb{R}$ : $|x - y| < 1$}. Is the relation emph{Q} reflexive, symmetric, transitive, irreflexive, and/or antisymmetric? Prove or disprove that each property holds.}}





section{fbox{6} textnormal{Prove that the collection {$A_{t}$ : $t in [0, 1)$}, where each of $A_{t}$ = {z + t : $z in mathbb{Z}$} is a partition of $mathbb{R}$.}}





section{fbox{7} textnormal{Write the addition and multiplication tables for $mathbb{Z}_6$, the set of equivalence classes for the relation $equiv$ (mod 6).}}





section{fbox{8} textnormal{Define a relation $leq$ on all words by agreeing that one word is $<$ another if an only if the first word comes before the second when arranged in alphabetical order. Show that this relation, called the emph{lexicographic order}, is a total order. (If you don't know why this order is called lexicographic, look up the definition of this word in a emph{lexicon}).}}





section{fbox{9} textnormal{Draw the Hasse diagram for the poset {{2}, {3}, {2, 3}, {4, 5}, {2, 3 , 4}, {4, 5, 6} ordered by set inclusion.}}





section{fbox{10} textnormal{Let emph{A} be a set and $subset$ be the ordering for emph{P}(emph{A}). Let emph{B} be a family of subsets of emph{A}. Prove that the least upper bound of emph{B} is $bigcup_{Xinemph{B}}X$ and the greatest lower bound of emph{B} is $bigcap_{Xinemph{B}}X$}}



end{document}

Welcome to TeX-SE! I would consider using an enumerate environment to make things fit (easily) on one page and using math mode for math symbols (instead of emph).

documentclass[a4paper, 12pt]{article}

usepackage{amsmath,amssymb,amsfonts}

newcommand{powerset}[1]{mathbb{P}(#1)}

DeclareMathOperator{Rng}{Rng}

DeclareMathOperator{Dom}{Dom}

begin{document}

begin{enumerate}renewcommand{labelenumi}{fbox{arabic{enumi}}}



item Let $A = {1, 2, 3}$. How many relations are there on $A$?





item Let $R$ be a relation from sets $A$ to $B$. Prove that $Rng(R^{-1}) =

Dom(R)$.





item Let $A = {a, b, 1, 2, 3}$.  Define $R = {(1, 1), (a, a), (a, b), (1,

2), (2, 1),$ $(a, 1), (a, 2)}$. Is this relation reflexive, symmetric,

transitive, irreflexive, and/or antisymmetric? Explain. Draw a digraph of $R$.





item Let $A = {a, b, 1, 2, 3}$. Let $mathsf{T}$ be an equivalence relation

on $A$ such that $amathsf{T}2$ and $bmathsf{T}3$. How many equivalence classes of $mathsf{T}$ are

possible? List the equivalence classes for all possibilities.





item Let $Q = {(x, y) in mathbb{R} times mathbb{R}~ :~|x - y| < 1}$. Is

the relation $Q$ reflexive, symmetric, transitive, irreflexive, and/or

antisymmetric? Prove or disprove that each property holds.





item Prove that the collection ${A_{t} : t in [0, 1)}$, where each of

$A_{t} = {z + t : z in mathbb{Z}}$ is a partition of $mathbb{R}$.





item Write the addition and multiplication tables for $mathbb{Z}_6$, the set

of equivalence classes for the relation $equiv$ ($pmod 6$).





item Define a relation $leq$ on all words by agreeing that one word is $<$

another if an only if the first word comes before the second when arranged in

alphabetical order. Show that this relation, called the emph{lexicographic order},

is a total order. (If you don't know why this order is called lexicographic,

look up the definition of this word in a emph{lexicon}).





item Draw the Hasse diagram for the poset ${{2}, {3}, {2, 3}, {4, 5},

{2, 3 , 4},$ ${4, 5, 6}}$ ordered by set inclusion.





item Let $A$ be a set and $subset$ be the ordering for

$P(A)$. Let $B$ be a family of subsets of $A$. Prove that

the least upper bound of $B$ is $bigcup_{Xin B}X$ and the greatest

lower bound of $B$ is $bigcap_{Xin B}X$.

end{enumerate}



end{document}