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Let $S$ be an infinite set and $S_1 \dots , S_n$ be sets such that $S_1 \cup S_2 \cup \dots \cup S_n = S$. Then

  1. at least one of the sets $S_i$ is a finite set
  2. not more than one of the sets $S_i$ can be finite
  3. at least one of the sets $S_i$ is an infinite
  4. not more than one of the sets $S_i$ can be infinite
  5. None of the above
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4 Answers

Best answer
36 votes
36 votes
  1. At least one of the set $S_{i}$ is a 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 sets into few sets (Some of which can be finite). This seems true, but I'm not able to prove it. Please Give a suitable counterexample here, if you think this is false.

          Ex-$: a^{\ast},$ this is infinite set. I can write it as $\{\}\cup \{a^{\ast}\},$ where $\{a^{\ast}\}$ is infinite.

  1. Not more than one of the sets can be finite. This is false.

         Ex $: a^{\ast}b^{\ast}\Rightarrow \{ab\} \cup  \{\} \cup \{{aa}^{+}{bb}^{+}\}.$

  1. At least one of the sets is Infinite. This must be True. As this is a finite union of sets, one of the sets must be infinite to make the whole thing infinite. True.
  2. Not more than one of the sets $S_i$ can be infinite. This is false. 

         Ex $: a^{\ast}b^{\ast} = \{a^{p}b^{q}|p=q\}\cup \{a^{m}b^{n}|m\neq n\}$ such that $p,q,m,n \geq 0.$

Answer C is surely true.

edited by
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0 votes

S= S1 U S2 U S3 U …..Sn

For S  to be infinite set at least one of sets S must be infinite, since if all Si were finite then S would also be finite.

 I hope my answer helps you a lot.

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