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{\Huge
21-128 problem sheet 1
}
Solutions to starred (*) exercises are due at the beginning of recitation on
\textbf{Thursday 10th September 2015}
Please submit answers to separate questions on separate sheets of paper.
\end{center}
\subsubsection*{Problem 1 --- 1.8}
In the morning section of a calculus course, $2$ of the $9$ women and $2$ of the $10$ men receive the grade of A. In the afternoon section, $6$ of the $9$ women and $9$ of the $14$ men receive an A. Verify that, in each section, a higher proportion of women than of men receive an A, but that, in the combined course, a lower proportion of women than men receive an A. Explain!
\subsubsection*{Problem 2 --- 1.15}
For what conditions on sets $A$ and $B$ does $A-B=B-A$ hold?
\subsubsection*{Problem 3 --- 1.22 *}
We have two identical glasses. Glass $1$ contains $x$ ounces of wine; glass $2$ contains $x$ ounces of water ($x \ge 1$). We remove $1$ ounce of wine from glass $1$ and add it to glass $2$. The wine and water in glass $2$ mix uniformly. We now remove $1$ ounce of liquid from glass $2$ and add it to glass $1$. Prove that the amount of water in glass $1$ is now the same as the amount of wine in glass $2$.
\subsubsection*{Problem 4 --- 1.27}
Determine the set of real solutions $x$ to the inequality
$$\left| \frac{x}{x+1} \right| \le 1$$
\subsubsection*{Problem 5 --- 1.29 *}
Let $x,y,z$ be nonnegative real numbers such that $y+z \ge 2$. Prove that $$(x+y+z)^2 \ge 4x+4yz$$
\subsubsection*{Problem 6 --- 1.32 *}
Assuming only arithmetic (not the quadratic formula or calculus), prove that
$$\{ x \in \mathbb{R} : x^2-2x-3 < 0 \} = \{ x \in \mathbb{R} : -1 < x < 3 \}$$
\subsubsection*{Problem 7 --- 1.36 *}
Let $S = [3] \times [3]$ (the Cartesian product of $\{ 1, 2, 3 \}$ with itself). Let $T$ be the set of ordered pairs $(x,y) \in \mathbb{Z} \times \mathbb{Z}$ such that $0 \le 3x+y-4 \le 8$. Prove that $S \subseteq T$. Does equality hold?
\subsubsection*{Problem 8 --- 1.42}
Let $A = \{ \text{January}, \text{February}, \dots, \text{December} \}$. Given $x \in A$, let $f(x)$ be the number of days in $x$. Does $f$ define a function from $A$ to $\mathbb{N}$?
\subsubsection*{Problem 9 --- 1.47 *}
Let $f : \mathbb{N} \times \mathbb{N} \to \mathbb{R}$ be defined by
$$f(a,b) = \frac{(a+1)(a+2b)}{2}$$
\begin{enumerate}[(a)]
\item Show that the image of $f$ is a subset of $\mathbb{N}$.
\item Determine exactly which natural numbers are elements of the image of $f$. (\textbf{Hint:} Formulate a hypothesis by trying values.)
\end{enumerate}
\subsubsection*{Problem 10 --- 1.54}
Let $S = \{ (x,y) \in \mathbb{R}^2 : y \le x \text{ and } x+3y \ge 8 \text{ and } x \le 8 \}$.
\begin{enumerate}[(a)]
\item Graph the set $S$.
\item Find the minimum value of $x+y$ such that $(x,y) \in S$. (\textbf{Hint:} On the graph from part (a), sketch the level sets of the function $f$ defined by $f(x,y)=x+y$.)
\end{enumerate}
\subsubsection*{Problem 11 *}
Let $r$ be a rational number and let $a$ and $b$ be irrational numbers. Which of the following numbers is necessarily irrational?
$$a+r \quad a+b \quad ar \quad ab \quad a^r \quad r^a \quad a^b$$
Prove your claims, either by proving that the number is irrational or by providing a counterexample. If you claim that a number is irrational, you should prove it.
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