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\title{\textbf{Homework 1}}

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\section*{Section 1.4}

\textbf{Exercise 7.}\vskip1ex
Prove $A \cap (B \cup C) = (A \cap B)\cup (A \cap C).$
\vskip1ex
\emph{Proof: } \vskip1ex
Your proof here. \hfill  (use this symbol to end proofs $\rightarrow$) $\square$
\vskip2ex
\textbf{Exercise 9.}\vskip1ex
Prove $(A \cap B)' = A' \cup B'.$
\vskip1ex
\emph{Proof: }
\vskip2ex

\textbf{Exercise 18(b, c).}\vskip1ex
Determine which of the following functions are one-to-one and which are onto. If the function is not onto, determine its range.
\begin{enumerate}
\item[(b)] $f: \Z \rightarrow \Z $ defined by $f(n) = n^2+3$
\item[(c)]$f: \R \rightarrow \R $ defined by $f(x) = \sin x$
\end{enumerate}
\emph{Solution:}
\vskip2ex

\textbf{Exercise 19.} \vskip1ex
Let $f: A \rightarrow B$ and $g: B \rightarrow C $ be invertible mappings. Show that $(g \circ f)^{-1}  = f^{-1} \circ g^{-1}$.
\vskip1ex
\emph{Proof:}
\vskip2ex

\textbf{Exercise 20(b).} \vskip1ex
Define a function $f: \N \rightarrow \N$ that is onto, but not one-to-one. \vskip1ex
\emph{Solution:}
\vskip2ex

\textbf{Exercise 21.} \vskip1ex
Prove that the relation defined on $\R^2$ by $(x_1, y_1) \sim (x_2, y_2)$ if $x_1^2+y_1^2=x_2^2+y_2^2$ is an equivalence relation.\vskip1ex
\emph{Proof:} \vskip2ex

\textbf{Exercise 22(c).} \vskip1ex
Let $f: A \rightarrow B$ and $g: B \rightarrow C$ be maps. Show that if $g \circ f$ is one-to-one then $f$ is one-to-one.
\emph{Proof:} \vskip2ex

%\textbf{Exercise 25(d).} \vskip1ex
%Determine whether or not the relation $m \sim n$ in ${\mathbb Z}$ if $m \equiv n \pmod{6}$ is an equivalence relation.
%If the relation is an equivalence relation, describe the partition given by it.
%If the relation is not an equivalence relation, state why it fails to be one.
%\vskip1ex
%\emph{Solution:} \vskip2ex

%\textbf{Exercise 26.} \vskip1ex
%Define a relation $\sim$ on $\R^2$ by stating that $(a,b) \sim (c,d)$ if and only if $a^2+b^2 \leq c^2+d^2$. Show that $\sim$ is reflexive and transitive but not symmetric.
%\vskip1ex
%\emph{Proof:} \vskip2ex

%\textbf{Exercise 28.} \vskip1ex
%Find the error in the following argument by providing a counterexample. ``The reflexive property is redundant in the axioms for an equivalence relation. If $x \sim y$,
%then $y \sim x$ by the symmetric property. Using the transitive property, we can deduce that $x \sim x$.
%\vskip1ex
%\emph{Solution:} \vskip2ex

%\textbf{Exercise 29. (Projective Real Line.)} \vskip1ex
%Define a relation on $\R^2 \setminus \{ (0,0) \}$ by letting $(x_1, y_1) \sim (x_2, y_2)$ if $(x_1, y_1)= (\lambda x_2, \lambda y_2)$ for some nonzero real number $\lambda$.
%Prove that $\sim$ is an equivalence relation on $\R^2 \setminus \{ (0,0) \}$. What are corresponding equivalence classes?
%(The equivalence relation defines the projective real line important in geometry.)
%\vskip1ex
%\emph{Proof:} 

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