To show that a set is countable, you need to show 1 to 1 correspondence, right? So to test if it is 1 to 1 and also onto. So for this example:

R is the set of real numbers. Let S = { x∈R | -3 < x < 0 }.
Show that the set S is not countable.

I can see this set is infinitely big. Also, all I see is that it IS 1 to 1 and onto because for example; 1 -> -2.999, 2 -> -2.990, 3 -> -2.900, ... Obviously, there is an infinite amount of numbers in between each of X values I selected. So you would shift all the numbers.

Z+ goes to infinite and the set goes from -3 to 0, infinitely.

But it says show that the set is not countable, so I am doing something wrong.

  • 1
    $\begingroup$ The same argument which proves that the real numbers are uncountable can also prove that real numbers between $-3$ and $0$ are uncountable. Simply generate an infinite list of real numbers between $-3$ and $0$ and show that there is always another one that is not in the list. $\endgroup$ – Edward Jiang Nov 10 '15 at 1:59
  • $\begingroup$ @EdwardJiang I see, I understand now, but in terms to help describe the set; does that make it 1 to 1 but not onto? $\endgroup$ – Nils_e Nov 10 '15 at 2:03

Hint : can You find a bijection between $\mathbb{R}$ and $S$?

  • $\begingroup$ Does 1 -> -2.999, 2 -> -2.990, 3 -> -2.900, ... count as a bijection between R and S? Even though you can add numbers in between each point. $\endgroup$ – Nils_e Nov 10 '15 at 1:40
  • $\begingroup$ @Nils_e I don't see any trend in your sequence. Where do you send 4 to, for instance? $\endgroup$ – BigbearZzz Nov 10 '15 at 1:49
  • $\begingroup$ @BigbearZzz 4 -> (to any number bigger than the previous one). so 4 can go to -> -2.8999. Is that the reason why it is not countable, because there is an infinite amount of choices for 4 to be? $\endgroup$ – Nils_e Nov 10 '15 at 1:52
  • $\begingroup$ So it is 1 to 1 but not onto? $\endgroup$ – Nils_e Nov 10 '15 at 2:02
  • $\begingroup$ You should use a function. $\endgroup$ – Jean-François Gagnon Nov 10 '15 at 5:01

We have that: $S=\{s\in \Bbb R: -3< s<0\}$

We can also see that: $S= \big\{ \boxed{?} \mathop{\big|} t \in\Bbb R, 0 < t < \infty\big\}$

So there is a bijection between $S$ and the positive reals, and further the positive reals are known to be uncountably infinite.

What goes in the box?


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