Let $L$ be a language.Define $\text{half(L)}$ to be the set of first halves of strings in $L,$ that is,$\{w|\text{for some x such that |x|=|w|,we have wx in L}\}.$For example,if $L=\{\in,0010,011,010110\}$ then $\text{half(L)}=\{\in,0,010\}.$ Notice that odd-length strings do not contribute to $\text{half(L)}$Prove that if $L$ is a regular language,so is $\text{half(L)}.$
