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Often when dealing with sets, we will have some understanding as to what “everything” is. Perhaps we are only concerned with natural numbers. In this case we would say that our universe is \(\N\text{.}\) Sometimes we denote this universe by \(\U\text{.}\) Given this context, we might wish to speak of all the elements which are not in a particular set. We say \(B\) is the complement of \(A\text{,}\) and write,

\begin{equation*} B = \bar A \end{equation*}

when \(B\) contains every element not contained in \(A\text{.}\) So, if our universe is \(\{1, 2,\ldots, 9, 10\}\text{,}\) and \(A = \{2, 3, 5, 7\}\text{,}\) then \(\bar A = \{1, 4, 6, 8, 9,10\}\text{.}\)

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