GO Classes DPP | TOC|COUNTABILITY

Answer 1

I. THE SET OF ALL CONTEXT-SENSITIVE LANGUAGES (CSLS).

• COUNTABLE. Every CSL can be generated by a CSL grammar or recognized by a Linear Bounded Automaton (LBA). An LBA is a TM with a finite description. The set of all finite descriptions is countable. Therefore, the set of CSLs is countable.
   

II. THE SET OF ALL FINITE SUBSETS OF $\mathbb{R}$.

• UNCOUNTABLE. Let $S_1$ be the set of all 1-element subsets of $\mathbb{R}$ (e.g., $\{1.5\},\{\pi\}, \ldots$ ). This set $S_1$ is in a one-to-one correspondence with $\mathbb{R}$ itself. Since $\mathbb{R}$ is uncountable, $S_1$ is uncountable. The total set of all finite subsets is a union containing this uncountable set, so it is uncountable.
   

III. THE SET OF ALL POSSIBLE JAVA PROGRAMS.

• COUNTABLE. A Java program is a finite-length string written using a finite alphabet (ASCII/Unicode). The set of all finite-length strings over a finite alphabet is countably infinite.