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\begin{document}
\title{Math 127 Homework}
\author{Mary Radcliffe}
\date{Due 5 April 2018}
\maketitle

Complete the following problems. Fully justify each response.


\begin{enumerate}

\item A number is called {\it algebraic} if it is the root of a polynomial $p(x)=a_nx^n+a_{n-1}x^{n-1}+\dots+a_1x+a_0$, where each $a_i\in \Z$. Let $\mathcal{A}$ denote the set of algebraic numbers.
\begin{enumerate}
\item Prove that $\Q\subseteq\mathcal{A}$.
\item Prove that the set of all algebraic numbers is countably infinite.

(Hint: First consider the possible roots of polynomials of degree $k$. Then use a union argument).
\end{enumerate}

\item \begin{enumerate}
\item Let $X$ be any set. Prove, using Cantor's Diagonalization Argument, that $|\mathcal{P}(X)|>|X|$.
\item Prove that $\mathcal{P}(X)$ is either finite or uncountably infinite.
\end{enumerate}

\item Let $f:X\to Y$ be a function. Define a relation $R$ on $X$ by $x_1 R x_2 \Leftrightarrow f(x_1)=f(x_2)$. Is this an equivalence relation? If so, prove it. If not, explain why not.

\end{enumerate}

\end{document}