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

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    $\begingroup$ The second one is "all ones or all zeroes", which doesn't include e.g. 0101010 $\endgroup$
    – vonbrand
    Feb 3 '14 at 4:41
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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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  (1*+0*)= 11.. or 00... but cannot produce 101010101
  (1+0)*= anything with 1 and 0

yeah: (1*+0*)* would be a different story.

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