Recall that a language $A $\backslash$subseteq $\backslash \{0,1\backslash \}^*$$ is $\backslash$emph$\{$prefix$-$free$\}$ if no element of $A$ is a prefix of any $\backslash$emph$\{$other$\}$ element of $A$$.$ Recall also that we proved the Kraft inequality, which says that, for every prefix$-$free language $A$$,$

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$$\backslash$displaystyle$\backslash$sum$\backslash$limits$\_\{$x $\backslash$in A$\}2^\{-|$x$|\}\backslash$leq $1$$$.$

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Define a prefix$-$free language $A$ to be $\backslash$emph$\{$maximal$\}$ if it is not a proper subset of any prefix$-$free language. Define a prefix$-$free language to be $\backslash$emph$\{$full$\}$ if

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$$\backslash$displaystyle$\backslash$sum$\backslash$limits$\_\{$x $\backslash$in A$\}2^\{-|$x$|\}=1$$$.$

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$\{\backslash$bf Problem $13.\}$ Prove that every full prefix$-$free language is maximal.$\backslash \backslash \backslash$

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