\documentclass[11pt]{exam}
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\title{Practice Exam 1}
\target{Math 157}

\begin{document}


\begin{exam}{\today}


\bp Use mathematical induction to prove that for all $n\in \N$,
$$0^3+1^3+2^3+\cdots +(n-1)^3<\frac{n^4}{4}\text{ and }\frac{n^4}{4}< 1^3+2^3+\cdots +(n-1)^3+n^3.$$
Use this result to compute the area of the region pictured below (the
vertical distance between the point $b$ units from $0$ is $b^3$).
\begin{center}
\includegraphics[scale=0.7]{practice_exam_pic.pdf}
\end{center}
\ep


\bp Prove that $\ds |x+y|\leq |x|+|y|$ for all $x,y\in \R$.
\ep


\bp Let $a,b\in \R$ with $a\neq 0$.  
Use the field axioms of $\R$ to carefully prove that if $ax+b=0$ then
$x=\frac{-b}{a}.$  Justify all your steps.
\ep


\bp True or False:  If $S$ is a nonempty subset of rational numbers
that is bounded above, then the least upper bound of $S$ is rational.  
\ep


\bp Prove or disprove:  For all propositions $p,q,r,s$, we have
$$((p\Rightarrow q) \Rightarrow r) \Rightarrow s \equiv ((p\wedge
q)\wedge r)\Rightarrow s.$$
\ep

\bp Let $X$ and $Y$ be sets.  Recall, for sets $X$ and $Y$, we define
the set $X\setminus Y$ to be $$X\setminus Y=\{x\in X \text{ satisfying
}x\notin Y\}.$$  Prove or disprove:
\ben
\item 
For all sets $A,B,C$ $$(A\setminus B)\cup C=(A\cup C) \setminus
(B\cup C).$$
\item 
For all sets $A,B,C$ $$A\setminus (B\cup C)=(A\setminus B) \cup
(A \setminus C).$$
\item 
For all sets $A,B,C$ $$A\setminus (B\cap C)=(A\setminus B) \cup
(A \setminus C).$$
\een
\ep



\bp Let $X$ and $Y$ be subsets of real numbers.  Define
the set $X-Y$ to be $$X-Y=\{x-y \text{ where }x \in X \text{ and }y\in Y\}.$$
% For example, if $X=\{
  % 2,3,5\}$ and $Y=\{0,1,2\}$ then $$X-Y=
  % \{2,3,5,1,2,4,0,1,3\}=\{0,1,2,3,4,5\}.$$

\ben
\item Suppose that $X,Y\subset \R$ and that $L$ is
  the least upper bound of $X$ and $M$ is the least upper bound of
  $Y$.  Prove or disprove:  the least upper bound of $X-Y$ is $L-M$.
\item Prove or disprove:  For any sets $X,Y\subset \R$, if $X\subset
  X-Y$ then $0\in Y$.
\een
\ep

\bp Use the order axioms of $\R$ to prove that if $x,y$ are positive
real numbers with $x<y$ then $x^2<y^2.$
\ep

\bp More induction:  use mathematical induction to prove that 
\ben
\item $\ds \sum_{k=1}^n =\frac{(n)(n+1)}{2}$ for all $n\in \N$.
\item $\ds n!>2^n$ for all natural numbers $n\geq 4.$
\een
\ep

\bp Negate the following propositions.  Decide whether the proposition
or its negation are true.
\ben
\item $\forall x \in \R \exists n\in \N (x<n)$
\item $\exists x \in \R \forall n\in \N (x>n)$
\item $\forall x>0 \exists n\in \N \left(
    \frac{1}{n}<x \right)$
\item $\forall x\in \R \forall y\in \R \left( x<y \Rightarrow \exists
    z\in \R (x<z<y)\right)$
\een
\ep

\end{exam}
\end{document}

