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\begin{document}
\title{Math 127 Homework}
\author{Mary Radcliffe}
\date{Due 24 January 2019}
\maketitle

Complete the following problems. Fully justify each response. You need only turn in those problems marked with a (*).

\begin{enumerate}

\item (*) Prove De Morgan's Laws as stated in Theorem 2 in the \href{http://math.cmu.edu/~mradclif/teaching/127S19/Notes/Propositional Logic.pdf}{Propositional Logic} notes.

\item Prove that $p\iff q$ is logically equivalent to $(p\ra q)\wedge (q\ra p)$.

\item (*) Write the following statement as a proposition using propositional variables and logical operators. You may assume that ``$x$ is odd'' is the logical negation of ``$x$ is even.''
\begin{quote}
Let $x$ be a positive integer. Then $x=2$, or $x$ is even and $x>2$, or $x$ is odd and prime, or $x$ is odd and composite.
\end{quote}

\item (*) Let $p, q, r$ be propositions. Show that \[\neg \left[ (p\wedge q)\vee (p\wedge(\neg r))\vee (q\wedge r)\right]\] is logically equivalent to \[\neg(q\wedge r)\wedge ((\neg p)\vee r).\]

\item Show that $(p\ra q)\vee (\neg q)$ is a tautology.

\item Let $x, y$ be rational numbers. Use the method of direct proof to show that $xy$, $x/y$, and $x-y$ are all rational numbers.

\item (*) Prove the following proposition: 
\begin{quote}Let $d, a, b, u, v$ be positive integers. If $a$ is divisible by $d$ and $b$ is divisible by $d$, then $au+bv$ is divisible by $d$.\end{quote}

\item (*) Follow the strategy of Example 4 in the \href{http://math.cmu.edu/~mradclif/teaching/127S19/Notes/Logic and Proof.pdf}{Logic and Proof} notes to prove the Quadratic formula:
\begin{quote}
A complex number $x$ is a solution to the equation $ax^2+bx+c=0$ if and only if \[x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}.\]
\end{quote}

\end{enumerate}
\end{document}