2 R is the set of real numbers Let S xR 3 x 0 Show tha

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

Please show work!

Solution

2.

Cantor\'s diagonal is a trick to show that given any list of reals, a real can be found that is not in the list.

First a few properties:

Cantor\'s diagonal is a clever solution to finding a number which satisfies these properties. The number which is the diagonal is transformed s.t. it doesn\'t share the first digit of the first number nor the second digit with the second and so on. Thus the number is unique to the list.

This is why Cantor\'s diagonal as a method proves the result that the reals are uncountable.