# Math 281 Exam 1

**Read** the problems. Read them again. Make sure to express your answers

in a ** clear and logical ** way. **Don't rush**!

1. (20 pts.)

Prove the following identity:

(5 pts. extra credit if you find a proof not by induction)

Hints and pointers:

(a) You can do a standard proof by induction here. Remember, a proof

by induction always consists of two steps . First, you need to assess

the validity of your formula for some basic case (e.g. n = 1). Second,

you assume the formula true for a generic n and from this you deduce

the validity of the formula for n + 1. Note that this often involves

manipulating parts of the formula for n + 1 so that you can reduce

it to the formula for n and some simple stuff .

(b) This identity lends itself also to a proof that does not involve
induction,

but for that a clever idea is required: find some A and B such

that

then start experimenting with this summation!

2. (20 pts.)

Prove that the set of real numbers which are roots of some quadratic

polynomial with integer coefficients is a countable set. In other words:

You must prove X is a countable set.

(5 pts. extra credit if you find two different proofs )

**Hints and pointers:**

Here there are two possible ways to prove this fact. The following questions

should lead you on the right track:

(a) How many quadratic polynomials with integer coefficients are there

(e.g. countably many or uncountably many?) For each polynomial,

how many roots can it have?

(b) How do you write down the roots of a quadratic equation ? (Quadratic

formula).

The two questions lead you in different directions, both of which work.

You certainly will want to make use of the theorems that we stated in

class:

•a countable union of countable sets is countable,

•a finite cartesian product of countable sets is countable,

•a countable union of finite sets is countable.

3. (20 pts.) Describe a bijection between an open disc in the plane and the

whole plane. In other words, let

You need to describe a function:

and show f is a bijective function.

**Hints and pointers:**

First o , you need to really understand what you are trying to do here!

You need to give me a function that associates to any point in the open

disc a point in the plane. This function must be into and onto!

This exercise is very similar to a problem in project 3. Note that there

you defined a bijection between (-1, 1) and all of R.

Also remember that you can, if you want, use polar coordinates on the

plane .

4. (25 pts.) If X is a finite set, prove that a function

is injective if and only if it is surjective. Show that this needs not be
true

if X is an infinite set.

Use this fact to prove that if p is prime every non-[0] element in Z/pZ has

a multiplicative inverse .

**Hints and pointers:**

First o , if you have two finite sets, and an injective function between

them, can you say something about which set has "more" elements? How

about if you have a surjective function?

The second part is trickier, and it requires being familiar with
multiplications

in Z/pZ. First try experimenting with multiplication tables for

various Z/pZ. Try to see what elements m have multiplicative inverses,

and if you can relate this to properties of m and n .

Then think of the function mult_{[n]} : Z/pZ -> Z/pZ. What does it mean

for n to have a multiplicative inverse in terms of this function ? But then

if you want to prove that mult_{[n]} is ****, maybe you can instead use the

previous part of the exercise to show that mult_{[n]} is ***.

5. (20 pts.)

Show that if z_{1} and z_{2} are two (complex) roots of 1 such that z_{2}-z_{1} = 1,

then they are 6-th roots of 1. In symbols :

if

and

then

**Caution:** be very careful about what you are
actually proving. I am not

asking you to prove that there exist two sixth roots of unity with that

property.

**Hint:** you might want to think about the geometric
representation of

roots of 1 (where do all roots of 1 live?). Also, take a complex number z.

Plot it on the plane. Then plot z-1. Where is z-1 with respect to z?

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