GO Classes DPP | TOC|COUNTABILITY

Answer 1

A. THE SET OF ALL LANGUAGES OVER $\Sigma$ IS COUNTABLY INFINITE.

• FALSE.
   
• The set of all strings over $\Sigma, \Sigma^*$, is countably infinite.
   
• A language is any subset of $\Sigma^*$.
   
• The set of all languages is the power set of $\Sigma^*, P\left(\Sigma^*\right)$.
   
• By Cantor's theorem, the power set of a countably infinite set is uncountable.

B. THE SET OF ALL COMPUTABLE FUNCTIONS $f: \mathbb{N} \rightarrow \mathbb{N}$ IS COUNTABLE.

• True.
   
• A function is computable if there exists a Turing Machine (TM) that computes it and halts on all inputs.
   
• Every TM can be finitely described by its rules and states, which can be encoded as a finitelength string (e.g., $\langle M\rangle)$.
   
• The set of all possible finite-length strings (and thus all possible TM encodings) is countably infinite.
   
• The set of all computable functions corresponds to a subset of these TM encodings (specifically, the ones that halt on all inputs).
   
• A subset of a countable set is also countable.

C. THE SET OF ALL FINITE SUBSETS OF $\mathbb{N}$ IS COUNTABLE.

• TRUE.
   
• Let $S_k$ be the set of all subsets of $\mathbb{N}$ with exactly $k$ elements.
   
• $S_0=\{\emptyset\}$ (Countable)
   
• $S_1=\{\{1\},\{2\}, \ldots\}$ (Countable)
   
• $S_k$ is countable for any finite $k$.
   
• The set of all finite subsets is the union $S_0 \cup S_1 \cup S_2 \cup \ldots$.
   
• This is a countable union of countable sets, which is countable.

D. THE SET OF ALL CONTEXT-FREE GRAMMARS (CFGS) OVER $\Sigma$ IS UNCOUNTABLE.

• FALSE.
   
• A CFG is defined by a 4-tuple $G=(V, T, P, S)$, where all components $(V, T, P, S)$ are finite.
   
• Because every component is finite, the entire grammar can be described by a finite-length string.
   
• The set of all possible finite-length strings is countable. Therefore, the set of all CFGs is countable.