Proving Countability of A: Real Numbers with Only 5 and 7 in Decimal Expansion

[giro]

Let A be the set of all real numbers in the interval [7,8) that have only 5 and 7 in their decimal expansion. A is defined by

A:={7.a₁a₂a₃|a_i ε {5,7} for all i ε [itex]\aleph[/itex]}

Prove A is countable.

 

[economicsnerd]

... Are you sure that's true?

Can you show that [itex]A[/itex] is the same size as [itex]\{S: \enspace S\subseteq \mathbb N\}[/itex]? Is the latter countable?

 

[tiny-tim]

hi giro! welcome to pf!

show us what you've tried and where you're stuck, and then we'll know how to help

 

[Mark44]

tiny-tim and economicsnerd,
When you see a post like this, that is pretty obviously a homework assignment, please use the Report button so that a mentor can deal with it.

 

[blue_raver22]

To prove that A is countable, we can use the method of diagonalization. This method involves constructing a list of all possible elements in A and showing that each element can be mapped to a unique natural number, thus proving that A is countable.

First, let us consider the decimal expansion of the numbers in A. Since A only contains numbers with decimal expansions consisting of only 5 and 7, we can represent each number in A as a sequence of 0s, 5s, and 7s. For example, 7.57 would be represented as (0, 5, 7, 0, 0, ...).

Next, we can construct a list of all possible sequences of 0s, 5s, and 7s, starting with sequences of length 1, then 2, and so on. This list would look like:

(0), (5), (7), (0, 0), (0, 5), (0, 7), (5, 0), (5, 5), (5, 7), (7, 0), (7, 5), (7, 7), (0, 0, 0), (0, 0, 5), (0, 0, 7), ...

We can see that every possible sequence of 0s, 5s, and 7s is included in this list. Now, to show that each element in A can be mapped to a unique natural number, we can use the following mapping:

(0) --> 1
(5) --> 2
(7) --> 3
(0, 0) --> 4
(0, 5) --> 5
(0, 7) --> 6
(5, 0) --> 7
(5, 5) --> 8
(5, 7) --> 9
(7, 0) --> 10
(7, 5) --> 11
(7, 7) --> 12
(0, 0, 0) --> 13
(0, 0, 5) --> 14
(0, 0, 7) --> 15
...

This mapping ensures that each element in A is assigned a unique natural number, and thus, A is countable.

In conclusion