Question

Determine whether each of these sets is countable or uncountable. For those that are countably in nite, exhibit a one-to-one correspondence between the set of positive integers and that set. (a) integers not divisible by 3. (b) the real numbers with decimal representations consisting of all 1s.

Answer 1

Answer:

a) Countably Infinite

b) Countably Infinite

Step-by-step explanation:

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

(a) integers not divisible by 3.

(b) the real numbers with decimal representations consisting of all 1s.

Proof -

a)

Let

A be the set of integers that are not divisible by 3

Now,

A is countably infinite.

The one-to one correspondence between the set to positive integers and A be -

f(n) = 3(n/4 - 1) + 1                if n is divisible by 4

     = 3(n-1)/4 + 2                 if n - 1 is divisible by 4 and n ≥ 1

     = -[ 3(n-2)/4 + 1]             if n - 2 is divisible by 4 and n ≥ 2

     = -[ 3(n-3)/4 + 2]             if n - 3 is divisible by 4 and n ≥ 3

b)

Let B be the set of the real numbers with decimal representations consisting of all 1s.

Now,

B is countably infinite.

The one-to one correspondence between the set to positive integers and B be -

When the function listed positive number, then we will list the corresponding negative number immediately after it before listing the next positive number.

Now,

The set of real number starts with 1, 11, 111, ... and the decimal representation consisting all 1s is -

1.1, 1.11, 1.111, 1.1111 .........(countable)

11.1, 11.11, 11.111, 11.1111 ........(countable)

111.1, 111.11, 111.111, 111.1111 ......(countable)

and so on

So,

There are countable infinite elements