Question

Consider the set S of real numbers in which the hundreth's digit in its decimal expansion is equal to 3. Is S a Finite, countably infinite, or uncountable set? Prove your claim.

Answer 1

Finite, is my guess, but have someone else look over it, cause I am not sure if my reading is correct

Answer 2

I would argue that this is an uncountable set.

Answer 3

A basic argument for that fact would be that you're taking "one-tenth" of the real numbers. Since it's an uncountable amount to start with, you must also end with an uncountable amount.

Answer 4

How do i go about proving that S is an uncountable set tho. !! like the steps involved and process !

Answer 5

Here's a sort of roundabout way to do it.

Multiply by 100 so that the 3's are in the one's place. Since we're working in the real numbers, we can just throw away the first two digits. Now subtract 3. This means that every number remaining is contained in the interval \[[0, 1]\]In fact, we contain every number in that set (with some repeats). This is well known to be uncountable, so S must be uncountable.

Answer 6

Thank you soo much.

Answer 7

You're welcome.