Question

Let AC (0,1) be the set of real numbers with a decimal expansion containing only Os, 2s, and 5s. For example, 2/9 = 0.222... € A and 0.2500525... E A, but 1/8 = 0.125 € A. Prove that A is uncountable.

Answer 1

In order to prove the first result we consider a subset of A, which consists of the real numbers with a decimal expansion containing only 0s and 2s, we name it as B. Then if we can show that B is uncountable, then obviously A becomes countable.

If possible let B be a countable set. Then there exists an one-one, onto map, f, from the set of natural numbers onto B.

$Let\;f(n)=x_n\in B.$

Now consider a number 'x' such that if the n-th term of the decimal expansion of xn is 0 then n-th term of the decimal expansion of 'x' is 2, and if the n-th term of the decimal expansion of xn is 2 then n-th term of the decimal expansion of 'x' is 0.

Then there exists no natural number 'n' such that xn=x. But it is clear that 'x' belongs to the set B. Hence the mapping 'f' cannot be onto. So B is uncountable and so is A.

For the second question let there be an binary operation 'f' with given properties.

Then as f(a,x)=x and f(x,a)=x for any 'x', putting x=b we have f(a,b)=b=f(b,a).

Again as f(b,x)=x and f(x,b)=x for any 'x', putting x=a we have f(b,a)=a=f(a,b). Which implies that a=b, which is not possible as 'a' and 'b' are distibct elements in the set A.