1
\documentclass[12pt]{article}
\usepackage{tipx}
\usepackage{textcomp}
\begin{document}
\title {Example Here}
\author{Example Here}
\date {Due October 12, 2016}
\maketitle


\textbf{Exercise 1} \\
Show that this implication is a tautology, by using a truth table: \\
\[ [(P \vee Q) \wedge (P \rightarrow R) \wedge (Q \rightarrow R)]         \rightarrow R \] \\

\pagebreak

\textbf{Exercise 2} \\

Show that the following is a tautology: \\
\[ (P \vee Q) \wedge (\neg{P} \vee R ) \rightarrow (Q \vee R) \] \\


\pagebreak

\textbf {Exercise 3} \\

a) Let x be a real number. Show that is \[x^2\] is irrational, then x is irrational.\\

b) Based on question a), can you say that "if x is irrational, it follows that \[x^2\] is irrational." ? \\ 

\textbf {Exercise 4} \\

Prove that a square of an integer ends with a 0, 1, 4, 5, 6, or 9. 
(Hint: let n = 10k + l, where l = 0, 1, ... , 9)    \\

\pagebreak

\textbf {Exercise 5} \\

Prove that if n is a positive integer, then n is even if and only if 5n + 6 is even.\\

\pagebreak

\textbf {Exercise 6} \\

Prove that either \[3.10^450 + 15\] or \[3.10^450 + 16\] is not a perfect square.
Is your proof constructive, or non-constructive? \\

\pagebreak

\textbf {Exercise 7} \\

Prove or disprove that if a and b are rational numbers, then \[a^b\] is also rational.\\

\textbf {Exercise 8} \\

Prove that at least one of the real numbers \[a_1, a_2, ... , a_n\] is greater than or equal to the average of these numbers. What kind of proof did you use?\\

\textbf {Exercise 9} \\
The proof below has been scrambled. Please put it back in the correct order.\\

\end{document}

My new question is how can I get x^2 without having it jump to a new line? I want it to stay in sentence. I tried ensuremath but whatever I'm doing doesn't work. Please test the code and see for yourself.

10
  • 2
    You need an \end{document} to make this code compile. The aforementioned content is really just warnings, so that shouldn't affect the compilation (or prevention of a resulting PDF). – Werner Oct 11 '16 at 15:26
  • 1
    Remove all the spurious \\ , and you'll have no warning. – Bernard Oct 11 '16 at 15:43
  • 2
    Use$x^2$ instead of \[x^2\]. The latter is for display one-line equations. – Bernard Oct 11 '16 at 15:46
  • 2
    Do you mean as in replacing \[a^b\] with $a^b$. \[...\] is used for displayed math and $...$ for inline math. – StefanH Oct 11 '16 at 15:47
  • 2
    $3.11^{450}$. Use of bracket is fundamental in (La)TeX for the arguments of a command. – Bernard Oct 11 '16 at 15:53
6

The document as posted produces

Underfull \hbox (badness 10000) in paragraph at lines 11--13


Underfull \hbox (badness 10000) in paragraph at lines 13--14

[1{/usr/local/texlive/2016/texmf-var/fonts/map/pdftex/updmap/pdftex.map}]
Underfull \hbox (badness 10000) in paragraph at lines 17--18


Underfull \hbox (badness 10000) in paragraph at lines 19--20


Underfull \hbox (badness 10000) in paragraph at lines 20--21

[2]
Underfull \hbox (badness 10000) in paragraph at lines 25--26


Underfull \hbox (badness 10000) in paragraph at lines 27--28


Underfull \hbox (badness 10000) in paragraph at lines 29--30

Underfull \hbox (badness 10000) in paragraph at lines 31--32


Underfull \hbox (badness 10000) in paragraph at lines 33--35

[3]
Underfull \hbox (badness 10000) in paragraph at lines 38--39


Underfull \hbox (badness 10000) in paragraph at lines 40--41

[4]
Underfull \hbox (badness 10000) in paragraph at lines 44--45

Underfull \hbox (badness 10000) in paragraph at lines 46--48

[5]
Underfull \hbox (badness 10000) in paragraph at lines 51--52


Underfull \hbox (badness 10000) in paragraph at lines 53--54


Underfull \hbox (badness 10000) in paragraph at lines 55--56


Underfull \hbox (badness 10000) in paragraph at lines 57--58


Underfull \hbox (badness 10000) in paragraph at lines 59--61

Since 10000 is as bad as TeX ever reports, then the log should be considered the only result, I didn't look at the pdf,

Fixing a few simple issues, removing all \\, using `` and '' not " using \item to enumerate items, using $ inline math, using {} to delimit superscripts, and using math for all instances of math not just where you need ^ so fonts are consistent removes all warnings and produces

enter image description here

\documentclass[12pt]{article}
\usepackage{enumitem}
\begin{document}
\title {Example Here}
\author{Example Here}
\date {Due October 12, 2016}
\maketitle


\begin{enumerate}[label=\textbf{Example \arabic*}]

\item 
Show that this implication is a tautology, by using a truth table:
\[ [(P \vee Q) \wedge (P \rightarrow R) \wedge (Q \rightarrow R)] \rightarrow R \]



\item

Show that the following is a tautology: 
\[ (P \vee Q) \wedge (\neg{P} \vee R ) \rightarrow (Q \vee R) \] 


\item

\begin{enumerate}[label=\alph*)]

\item Let $x$ be a real number. Show that is $x^2$ is irrational, then $x$ is irrational.

\item Based on question a), can you say that ``if $x$ is irrational, it follows that $x^2$ is irrational.''? 
\end{enumerate}

\item

Prove that a square of an integer ends with a 0, 1, 4, 5, 6, or 9. 
(Hint: let n = 10k + l, where l = 0, 1, ... , 9)    

\item

Prove that if $n$ is a positive integer, then $n$ is even if and only if $5n + 6$ is even.

\item

Prove that either $3.10^{450} + 15$ or $3.10^{450} + 16$ is not a perfect square.
Is your proof constructive, or non-constructive? 

\item
Prove or disprove that if $a$ and $b$ are rational numbers, then $a^b$ is also rational.

\item

Prove that at least one of the real numbers $a_1, a_2, \ldots , a_n$ is greater than or equal to the average of these numbers. What kind of proof did you use?

\item
The proof below has been scrambled. Please put it back in the correct order.

\end{enumerate}

\end{document}

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