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How do I present the complement using set notation?

I guess it has to be shows with universal set - {aa,bb} but I do not know how to represent the universal set in terms of set notation. Since, the strings of the universal set can be anything over alphabets {a,b}. So, how to represent it? I guess something like this might work if we are going from outside to inside {{{a,b}*}*}. Any help is appreciated.

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The universe here is the set of all strings over the alphabet $\{a,b\}$ and is usually denoted by $\{a,b\}^*$.

You have way too many brackets.

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    $\begingroup$ $\Sigma^*$ is also popular. $\endgroup$ – adrianN Aug 22 '16 at 13:16
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    $\begingroup$ @adrianN Yes, when the alphabet is $\Sigma$. Here it is assumed that $\Sigma = \{a,b\}$. $\endgroup$ – Hendrik Jan Aug 22 '16 at 13:46
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How do I present the complement using set notation?

There is no universally standard way of representing a set complement. But there are a few commonly used notations. If $S$ is a set, then its complement can be denoted by:

  1. $S^c$. My personal preference.

  2. $\overline S$. I've seen this in a few textbooks but I don't like it because I first learned this notation to mean the mathematical closure of a set.

  3. $U \setminus S \quad $ (or $U-S$)$\quad$ if the universal set $U$ is known.

  4. $\lnot S$

Because there is no universal standard notation for set complement, it's best to pick whichever common notation you like (if one hasn't already been established in the context) and specify what it means the first time you use it.

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