[Solved]: Difference between 1* + 0* and (1 + 0)*

Problem Detail: 

I know that (1 + 0)* is the set of all bit strings; but isn't 1* + 0* the same thing?

Asked By : O.A.

Answered By : Yuval Filmus

The set $1^*+0^*$ is composed of two parts: $1^*$ and $0^*$. The first part, $1^*$, is all strings composed entirely of $1$s. The second part, $0^*$, is all strings composed entirely of $0$s. In contrast, $(1+0)^*$ is all strings composed of $0$s and $1$s. Can you now think of a string in $(1+0)^*$ but not in $1^*+0^*$?

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Question Source : http://cs.stackexchange.com/questions/20235