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\title{MATH 3040: Homework 7}
\author{}
\date{October 11, 2017}

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All answers must be in complete sentences. All assertions must be proven, per the student contract. Remember that this is a writing class, so while correctness and thoroughness are most important, \textit{style} matters as well. 


\bigskip
\noindent
\textbf{1. (Practice in Posing Questions)} What is an interesting open-ended question you could ask, which is related to the material that we covered in class? What is a concrete mathematical question that could be used to approach an answer to the question? 


\bigskip
\noindent
\textbf{2. (Images) } Let $f: A \to B$ be a function. Let $C$ and $D$ be subsets of $A$.
\begin{enumerate}
  \item{Prove that $f(C \cup D) = f(C) \cup f(D)$.}
  \item{Prove that $f(C \cap D) \subset f(C) \cap f(D)$.}
  \item{Give an explicit example of a function $f$ in which $f(C \cap D)$ is a \textit{proper} subset of $f(C) \cap f(D)$ (i.e. where (b) holds, but where $f(C \cap D) \neq f(C) \cap f(D)$).}	
\end{enumerate}



\bigskip
\noindent
\textbf{3. (A cool bijection)} Prove that the function $f: \NN \times \NN \to \NN$ defined by
\[
f(m,n) = 2^{m-1}(2n -1)
\]
is a bijection.

\bigskip
\noindent
\textbf{4. (Properties about sets, functions, and cardinality)} Prove each of the following statements.
\begin{enumerate}
  % a
  \item{Let $A$ and $B$ be nonempty finite sets. The following statements are equivalent:
    \begin{itemize}
      \item[(i)]{There is an injection from $A$ to $B$.}
      \item[(ii)]{There is a surjection from $B$ to $A$.}
      \item[(iii)]{$|A| \leq |B|$.}	
    \end{itemize}

  }	
  
  % b
  \item{Let $A$ and $B$ be nonempty sets such that $|A| = |B|$, and let $f: A \to B$ be a function. If $f$ is injective or surjective, then $f$ is a bijection.}
  
  % c
  \item{Let $A$ and $B$ be nonempty finite sets. Show that at least one of the following statements is always true: (1) There is an injection from $A$ to $B$; (2) There is a surjection from $A$ to $B$. }
\end{enumerate}






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\noindent
The following question is to be done as a group.

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\noindent
\textbf{5. (Groupwork)} A familiar way to get functions $f: \RR \to \RR$ is to define $f(x)$ to be a polynomial, such as $f(x) = x^3$. These are more or less the subject of a first course in calculus, if we assume that $f$ is continuous and differentiable (and all polynomials are both). We now want to study the properties of these polynomial functions in the context of a function between sets.
\begin{enumerate}
  \item{Prove that every linear polynomial function $f: \RR \to \RR$ is a bijection. (A linear polynomial is a degree-one polynomial, i.e. $f(x) = ax + b$ for $a, b \in \RR$.)}
  
  \item{Prove that no quadratic polynomial function $f: \RR \to \RR$ is a bijection. (A quadratic polynomial is a degree-two polynomial, i.e. $f(x) = ax^2 + bx + c$ for $a, b, c \in \RR$.)}
  
  \item{(This question requires a little calculus.) Find a simple ``if and only if" statement in terms of the coefficients $a, b, c, d \in \RR$ that characterizes when a cubic polynomial function $f: \RR \to \RR$ given by
  \[
  f(x) = ax^3 + bx^2 + cx + d
  \]
  is a bijection, and prove that your characterization is correct. (This question can be tricky. Play around with some examples and see if you can find any patterns.)
  }	
\end{enumerate}
 
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