@Debapaul

As S is singleton set, consider S= {∊}

S* ={∊} which is a finite set.

Hence S1 is false as it says S* will always be countably infinite set.

However S2 is false as it says S* will always be finite. Consider S= {a} then S* ={ ∊, a, aa, aaa,...}

Dont consider S to be null set as it is mentioned that S is a singleton set