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I have the following 4 claims about languages. I need to state whether each is true/false and provide a short justification.

  1. $\forall L : (L^+)^* = L^*$
  2. $\forall L_1, L_2 : (L_1 = L_2 \iff L^*_1 = L^*_2)$
  3. $\forall L_1, L_2 : L_1 L_1^* L_2 \subseteq L_1^* L_2$
  4. $\forall L_1, L_2 : L_1^* L_2 \subseteq L_1 L_1^* L_2$

Progress so far:

First I begin with a selection of definitions provided by my reference literature:

  • Alphabet "$\sum$": A non empty finite set.
  • Symbol: An element from an alphabet.
  • Word: a finite sequence of symbols from an alphabet.
  • Empty word ($\varepsilon$): A sequence consisting of 0 symbols.
  • Language ($L$): A subset from $\sum^*$
  • $L^* = \bigcup_{i \geq 0} L^i$
  • $L^+ = \bigcup_{i \geq 1} L^i$

Attempts at solutions:

  1. I have found a solution which claims true because $L^* = L^+ \cup \{ \varepsilon \}$. Perhaps someone could elaborate here? I interpret the question to mean that we first we take the union of all words $\geq 1$ and then the union of all worlds $\geq 0$, i.e I first compute the term inside the brackets before I compute the outer $*$. Am I making an error here?
  2. I have a solution which claims that this is false by providing the counter example $\forall w \in L_1 : w \in L_2 \land \forall w \in L_2 : w \in L_1, L_1 = \{ a , b \}, L_2 = \{ a , b, a, b \}$. I thought that it was against the rules to define duplicate items in a set? I don't understand the idea here.

For the remaining solutions I have not made any progress.

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$L \subseteq L^+$ so $L^\ast \subseteq (L^+)^\ast$.

The reverse is also true: if $x$ is a word in $(L^+)^\ast$, it is either equal to $\varepsilon$ which is also in $L^\ast$, or $x= w_1w_2 \cdots w_n$ where all $w_i$ are from $L^+$, so themselves concatenations of some finite number of words from $L$. So that makes $x$ of the same form, hence in $L^\ast$.

Check that $L_1 = \{0,1,00\}$ and $L_2 = \{0,1,10\}$ obey $L_1^\ast = L_2^\ast$.

3 is true as $L_1L_1^\ast \subseteq L_1^\ast$ already. Quite trivial to check.

  1. $L_1 =\{0\}$ and $L_2 = \{1\}$ is a counterexample as $1$ is in the left language but not in the right one.
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