Just doing some work on Finite and infinite languages. And came across some statements I know the answer to but not sure how to explain why.

  1. There are finitely many finite languages. -This is false right? Since there are technically infinitely many finite languages

  2. Union of any two languages over alphabet (1,0) is regular. -False right? Since you can take a (non-reg U non-reg = non-reg)

  3. Single state NFA can recognize only finite languages -False, but I have no idea how to explain why...

  1. False. There are infinitely many finite language. Just think of {0,1}*. This is a infinite set of finitely long strings. or say, it is union of infinite number of finite languages over {0,1} which have strings of finite length.

  2. False. Union of two non regular languages MAY BE regular but it not the case always. This answer explains it well.

  3. False. Lets have a single state NFA having state q0. let 'd' be the transition function. d(q0, a) = q0 and since this is single state NFA, q0 is also final state. This NFA will accept all strings having only 'a'. i.e. {a,aa,aaa,aaaa...} The cardinality of this set is infinite, hence it can accept infinite languages too.

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    $\begingroup$ A union of two languages can be regular even if it is not the case that they are both regular. $\endgroup$ – Yuval Filmus Oct 10 '19 at 23:04
  • $\begingroup$ @YuvalFilmus I have edited the second point. But I'm not sure if the first one is totally correct. Can you please have a look at it? $\endgroup$ – Breakpoint Oct 11 '19 at 8:27

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