# Problem 1: Countability

Prove that the language \(L = \Sigma^*\) where \(\Sigma = \{\, 0, 1 \,\}\) is countably infinite.

(

*Hint*: simply declaring that the mapping from \(\Sigma^*\) to \(\mathbb{N}\) is the binary interpretation of \(w \in \Sigma^*\) is insufficient since \(0\) and \(000\) are distinct strings but both represent the value zero. The resulting mapping would, therefore, not be a bijection!}Generalize your construction to any ( ) of finite size.

Use this fact to argue that the set of possible Java programs is countably infinite.

# Problem 2: Hey, Wait a Second

Consider the following proof that \(\mathbb{N}\) is uncountable.

*Proof*. Assume that \(\mathbb{N}\) is countable. Then there is a bijection \(f\) that covers every natural numbers in \(\mathbb{N}\). Construct the natural number \(n\) where the \(i\)th digit of \(n\) is the \(i\)th digit of the \(i\)th natural number in the bijection (*i.e.*, \(f(i)\)) plus one mod 10 (so that it is a decimal digit). That is, if \(k\) is the \(i\)th digit of the \(i\)th natural number, then the \(i\)th digit of \(n\) is given by \(k + 1 \mod 10\). \(n\) is a valid natural number and by construction, \(n\) differs from every natural number in the bijection by one digit. Therefore, \(n\) cannot be in the bijection and therefore our assumption that such a bijection exists is incorrect. Thus, \(\mathbb{N}\) is uncountable.

However, we already know that \(\mathbb{N}\) is countable (the bijection is the identity function). What is wrong with this proof?

(*Hint*: think about the assumptions latent in Cantor’s diagonalization argument that the reals are uncountable. What specifically is different in this case?}

# Problem 3: Running the (Diagonal) Line

Consider the following language:

\[ \mathsf{HALT}_\mathsf{TM} = \{\, (M, w) \mid \text{Turing machine \( M \) halts on input \( w \)} \,\}. \]

By “halts”, we mean that the Turing machine either accepts or rejects. Adapt the diagonalization proof that \(A_\mathsf{TM}\) is undecidable from the end of chapter 4.2 to show that \(\mathsf{HALT}_\mathsf{TM}\) is undecidable. In a few sentences, describe how the diagonalization matrix figure 4.21 changes when adapted to your new proof.