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+4 votes
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How $\phi^*=\epsilon$?
asked in Theory of Computation by Boss (14.4k points) | 406 views

1 Answer

+9 votes
Best answer
$\phi = \{\}$, which is empty set. Now by definition of *, it repeats the content of a set ZERO or more times. And anything repeated zero time is $\epsilon$. And nothing repeated 1 or more times is nothing.

So, $\phi^*$ generates  the language $\{\epsilon\}$ whose regular expression is $\epsilon$. Hence,
$$\phi^* = \epsilon$$

It can be also shown as:

$a^* \text{ generates } \{\epsilon, a, aa, aaa, ....\}$
$\phi^* \text{ generates } \{\epsilon\}$
answered by Veteran (363k points)
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