A) at least one of the set $S_{i}$ is Finite Set. Well it is not said that$S_{1},S_{2}\ldots S_{n}$ whether they are finite or infinite. It is possible to break down infinite set into few sets (Some of which can be finite). This seems True, but I'm not able to prove it. Please Give suitable counterexample here, if you think this is false.
Ex-$: a^{*},\text{this is infinite set. I can write it as} \{\}\cup \{a^{*}\},$ where $\{a^{*}\}$ is infinite.
B) Not more than one of set can be finite -> This is false.
ex $: a^{*}b^{*}\Rightarrow \{ab\} \cup \{\} \cup \{{aa}^{+}{bb}^{+}\}.$
C) at least one of the sets is Infinite -> This must be True. As this is finite union of sets, one of set must be infinite to make whole thing infinite. True.
D) not more than one of the sets $S_i$ can be infinite. This is false.
Ex $: a^{*}b^{*} = \{a^{p}b^{q}|p=q\}\cup \{a^{m}b^{n}|m\neq n\}$ such that $p,q,m,n \geq 0.$
Answer -> C is surely true.
ESi