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\begin{document}
\title{Math 127 Homework}
\author{Mary Radcliffe}
\date{Due 22 February 2018}
\maketitle

Complete the following problems. Fully justify each response.

\begin{enumerate}

\item Let $X, Y, Z$ be sets, with $X,Y\subseteq Z$. Prove that \[\left[ (Z\ba X)\cap (Z\ba Y)\right] \cup (X\ba Y) = Z\ba Y.\]

\item Let $X$ be a set. Prove that $X\times \emptyset = \emptyset$.

\item Let $X, Y, Z$ be sets. Is it true that $X\times (Y\times Z) = (X\times Y)\times Z$? Explain your answer with a proof or a counterexample.

\item For each of the following subsets $G$ of $X\times Y$, determine if the subset represents the graph of a function from $X\to Y$. If so, specify the function.
\begin{enumerate}
\item $X=\R$, $Y=\R$, $G = \{(x, x+1)\ |\ x\in \R\}$.
\item $X=\R$, $y=\R$, $G=\{(x^2, x)\ | \ x\in \R\}$.
\item $X=\R^+$, $y=\R^+$, $G=\{(x^2, x)\ | \ x\in \R^+\}$.
\item $X=\Q$, $y=\Q$, $G=\{(x, y\ | \ x, y\in \Q\hbox{ and }xy=1\}$.
\end{enumerate}

\item Which of the following function specifications are well-defined? If one is not well-defined, determine a modification to the specification that would rectify the issue.
\begin{enumerate}
\item $g:\Q\to\Q$ defined by $g(x)(x+1)=2$.
\item $f:\Q\to\R$ defined by $f(x)(x+\pi)=1$.
\item $h:\R\to\R$ defined by $h(x)=\sqrt{x}$.
\item $\ell:\C\to \C$ defined by $\ell(x)=\sqrt{x}$.
\end{enumerate}

\item Let $f, g, h, \ell: \R\to \R$ be functions with the following specifications:
\[ f(x)=x+2;\quad g(x)=x^2; \quad h(x)=\frac{1}{x^2+1}; \quad \ell(x)=-x.\]
Write a specification, via a single equation, for each of the following:
\begin{enumerate}
\item $f\circ g$.
\item $g\circ f$.
\item $f\circ(g\circ(h\circ \ell))$.
\item $(f\circ g)\circ (h\circ \ell)$.
\end{enumerate}

\end{enumerate}

\end{document}