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I have to provide a DFA for the following language: $L=\{w|w$ is any string not in $a^* \cup b^*\}$

what does $a^* \cup b^*$ mean?

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    $a^{\ast}$ is the language of all strings that contain only $a$. Same for $b^{\ast}$. Union of them is the language of all strings that contain only $a$ or only $b$. – 2017-01-27
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    so it can be any string with a or b. – 2017-01-27
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    The language $L$ in your question is the language of strings that contain both $a$ and $b$. $ab\in L$, but $aaa\notin L$ and $b\notin L$. – 2017-01-27
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    a* can contain the empty character as well right? – 2017-01-27
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    I think that the problem is reduced to $L=\{w|w$ is any string not in $a^*$ or $b^*\}$ – 2017-01-27
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    Indeed, any number of a's, including 0. – 2017-01-27
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    so $b\in L$ right?, but $ba \notin L$ – 2017-01-27
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    Let us [continue this discussion in chat](http://chat.stackexchange.com/rooms/52566/discussion-between-galc127-and-themathnoob). – 2017-01-27

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$\cup$ is the usual set union symbol. $a^*$ is the set $\{ \epsilon, a, aa, aaa, \dots\}$; $b^*$ likewise is $\{ \epsilon, b, bb, bbb, \dots\}$. The union is $\{ \epsilon, a, aa, aaa, \dots, b, bb, bbb, \dots\}$.