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If $A$ is any language,let $A_{\frac{1}{2}-}$ be the set of all first halves of strings  in $A$ so that $A_{\frac{1}{2}-}=\{\text{x|for some y,|x|=|y| and xy$\in$A\}}.$ Show that if $A$ is regular,then so is $A_{\frac{1}{2}-}.$
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