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.

Create a function (prob3_6) that will do the following: input a positive scalar integer x. if x is odd, multiply it by 3 and add 1. if the given x is even, divide it by 2. repeat this rule on the new value until you get 1, if ever. your program will output how many operations it had to perform to get to 1 and the largest number along the way. for example, start with the number 3: because 3 is odd, we multiply by 3 and add 1 giving us 10. 10 is even so we divide it by 2, giving us 5. 5 is odd so we multiply by 3 and add one, giving us 16. we divide 16 (even) by two giving 8. we divide 8 (even) by two giving 4. we divide 4 (even) by two giving 2. we divide 2 (even) by 2 to give us 1. once we have one, we stop. this example took seven operations to get to one. the largest number we had along the way was 16. every value of n that anyone has ever checked eventually leads to 1, but it is an open mathematical problem (known as the collatz conjectureopens in new tab) whether every value of n eventually leads to 1. your program should include a while loop and an if-statement.