Define the prefix operator for a string w as prefix$($w$)=\{$w$1$ wk $|$ k $|$w$|\}.$ This operator generates all

contiguous substrings of w containing w$1.$ For example, if w $=0100,$ then prefix$($w$)=\{$$,0,01,010,0100\}.$

Let the prefix of a language L be defined as prefix$($L$)=$ Uw in L prefix$($w$)$; i$.$e$.,$ all prefixes of all strings in

L$.$ Are the context$-$free languages closed under prefix? Answer yes or no$,$ then prove your answer.