Problem 1E

Determine whether each of these sets is finite, countably infinite, or uncountable. For those that are countably infinite, exhibit a one-to-one correspondence between the set of positive integers and that set.

a) the negative integers

b) the even integers

c) the integers less than 100

d) the real numbers between 0 and

e) the positive integers less than 1.000.000,000

f) the integers that are multiples of 7

Step 1</p>

We have to check the following sets are finite, countably infinite, or uncountable.

Step 2</p>

a)The negative integers

Set of negative integers is countably infinite.

Then we can define a one -one function from set of positive integers to set of negative integers as,

is defined as f(z) = -z

Step 3</p>

b) The even integers

Let E =

E = { …-4,-2,2,4,6,8,...}

The set of all even integers are countably infinite.

Define a one-one function from the set of positive integers to the set of even integers as

as f ( z ) = +- 2z

Step 4</p>

c)The integers less than 100.

Let A = { z in Z : z < 100}

A = { …-2,-1,0,1,2,...,100}

Clearly A is countably infinite.

We define a one-one function from set of positive integers to A as

is defined as f(a) =100-a