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How $\phi^*=\epsilon$?
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$\phi = \{\}$, 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:

$a^* \text{ generates } \{\epsilon, a, aa, aaa, ....\}$
$\phi^* \text{ generates } \{\epsilon\}$
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