Homework I

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The first homework is due Thursday, September 12th, recitation time. If you cannot make it to the recitation, please let me know beforehand (by email or in person after the class on Tuesday).

The homework consists of the following five problems from the first chapter:

P.S: For Exercise 1.7 please use the definitions from Note 1.5.

For Exercise 1.4, you may use the following reasonings.

Similarly (P is a sufficient condition for Q) is false only when P is true but Q is false.

You are allowed to decide when (P unless Q) is false, on your own. But please explain your reasoning.
Here is my reasoning for "unless": (P unless Q) is false only when Q is not true and P is not true. For me this sounds like I am promising to do P as long as Q does not hold true. In other cases I don't promise anything. So I will be lying only when Q does not hold true and I don't keep my promise.

For Exercise 2.9, you may use the following definitions:

$\displaystyle lim_{x\to a} f(x) = L$

$\epsilon$

$\delta$

$0<\vert x-a\vert<\delta$

$\displaystyle \vert f(x)-L\vert<\epsilon \; .$

$\epsilon$

$\delta$

$\vert x-a\vert<\delta$

$\displaystyle \vert f(x)-f(a)\vert<\epsilon \;$

$\displaystyle lim_{x\to a} f(x) = f(a)$

Shortcut: If you want to use a shortcut for this problem, what you can do is to define the set of positive real numbers giving the notation you like. For example you might use ${\bf R}_{>0}$ or $(0,\infty)$ as a notation.

Good luck!