6.1.4. Difference between Finite and Infinite Length Binary Strings

Try these first

1. 1.

   What is the primary difference between finite and infinite length binary strings?

   Hint

   Think about whether the strings end or not.

2. 2.

   Provide an example of a finite length binary string.

   Hint

   Remember these strings have to end.

3. 3.

   Which of the following is a finite length binary string?

   • 000
   • 000...
   • 111...

   Hint

   Consider which strings have a termination.

4. 4.

   True or False: The set of infinite binary strings is countable.

   • True
   • False

   Hint

   Think about Cantor's argument.

5. 5.

   Create a new string that cannot be found in any enumeration of infinite binary strings. How does it relate to Cantor's argument?

   Hint

   Remember how Cantor's diagonalization creates contradictions.

6. 6.

   Analyze the implications of Cantor's argument on the nature of real numbers. Why do real numbers contain both rational and irrational sets?

   Hint

   Think about how decimals behave compared to whole numbers.