4. Consider the following recursive definition: Basis: 0,1,2,7,9 s Recursive step: If Ox e S, then 10x e S, 20% e S, and 70x e S. If lx e S, then 01x ES, 21x e S, and 71x e S. If 7x e S, then 07x e S, 17x e S, and 27x e S. a. What is the smallest possible alphabet for S? (3 points) b. What is the maximum number of 2 symbols that a string in S can have? (3 points) c. If a string in S contains a 2 symbol, where will that 2 symbol be located in the string? (3 points) d. Why is it impossible for the string 110 to belong to S? (3 points) e. Show that 2017 e S. (10 points) 5. Consider the recursive definition of the natural numbers as discussed in class. a. Basis: 0 EN b. Recursive step: if n e N, then s(n) EN Give a recursive definition of D, the set of natural numbers that are divisible by 2 and divisible by 3. You must use successor notation properly instead of addition throughout the definition. (15 points) 6. Define T* properly using mathematical notation (i.e., with union, set concatenation, other known sets, etc.). (4 points) 4. Consider the following recursive definition: Basis: 0,1,2,7,9 s Recursive step: If Ox e S, then 10x e S, 20% e S, and 70x e S. If lx e S, then 01x ES, 21x e S, and 71x e S. If 7x e S, then 07x e S, 17x e S, and 27x e S. a. What is the smallest possible alphabet for S? (3 points) b. What is the maximum number of 2 symbols that a string in S can have? (3 points) c. If a string in S contains a 2 symbol, where will that 2 symbol be located in the string? (3 points) d. Why is it impossible for the string 110 to belong to S? (3 points) e. Show that 2017 e S. (10 points) 5. Consider the recursive definition of the natural numbers as discussed in class. a. Basis: 0 EN b. Recursive step: if n e N, then s(n) EN Give a recursive definition of D, the set of natural numbers that are divisible by 2 and divisible by 3. You must use successor notation properly instead of addition throughout the definition. (15 points) 6. Define T* properly using mathematical notation (i.e., with union, set concatenation, other known sets, etc.). (4 points)