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Show that there is a language a ⚆ {0, 1} â— with the following properties: 1. for all x â a, |x| ≤ 5. 2. no dfa with fewer than 9 states recognizes a. hint: you don’t have to define a explicitly; just show that it has to exist. count the number of languages satisfying (1) and the number of dfas satisfying (2), and argue that there aren’t enough dfas to recognize all those languages. to count the number of languages satisfying (1), think about writing down all the strings of length at most 5, and then to define such a language, you have to make a binary decision for each string about whether to include it in the language or not. how many ways are there to make these choices? to count the number of dfas satisfying (2), consider that a dfa behaves identically even if you rename all the states, so you can assume without loss of generality that any dfa with k states has the state set {q1, q2, . . , qk}. now think about how to count how many ways there are to choose the other four parts of the dfa.

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Write the command that can be used to answer the following questions. (hint: try each out on the system to check your results.) a. find all files on the system that have the word test" as part of their filename. b. search the path variable for the pathname to the awk command. c. find all files in the /usr directory and subdirectories that are larger than 50 kilobytes in size. d. find all files in the /usr directory and subdirectories that are less than 70 kilobytes in size. e. find all files in the / directory and subdirectories that are symbolic links. f. find all files in the /var directory and subdirectories that were accessed less than 60 minutes ago. g. find all files in the /var directory and subdirectories that were accessed less than six days ago. h. find all files in the /home directory and subdirectories that are empty. i. find all files in the /etc directory and subdirectories that are owned by the group bin."