need help in Questions 3 ,9,11 ?

Export P 2. Using induction on i, prove that (WR)-(w()R for any string bu and all i 20. 3. Prove, using induction on the lengthofthe string, that (WR)R- forallstringswer 4. Lex-Iaa. bbl and Y-1, b, ab). Create Edit PDF a) List e strings in the set XY. b) How many strings of length 6 are there in X* c) List the strings in the set Y" of length three or less. d) List the strings in the set XY of length four or less. Comme Combin 5. Let L be the set of strings over {a, generated by the recursive definition i) Basis: bEL i) Recursive step: if u is in L then ub e L, uab el, and uba eL, and bua eL ii) Closure: a string v is in L only if it can be oblained from the basis by a finite Organiz Protect number of iterations of the recursive step. a) List the elements in the sets Lo. L1, and L2 b) Is the string bbaaba in L? If so, trace how it is produced. If not, explain why not. c) Is the string bbaaaabb in L? If so, trace how it is produced. If not, explain why not. Redact Fill &si 6. Give a recursive definition of the set of strings over la, b) that contain at least one b and have an even numberof a'sbefore the first b.For cxample, bab, aab, andaaaabababab are in the set, whilc aa, abb are not. Send fol 7. Give a recursive definition of the set(a' b10 i-j 2]. Send & 8. Give a recursive definition of the set of strings over la, b] that contain twice as many 9. Prove that every string in the language defined in Example 2.2.1 has even length. The 10. Prove that every string in the language defined in Example 2.2.2 has at least as many a's a's as b's +More To proof is by induction oa the recursive gencration of the strings. as b's. Let na ) denote the number of a's in the string u andn() denote the number of b's in . The inductive proof should establish the inequality ngu). 11. Let L be the language over la, b} generated by the recursive definition i) Basis: e L il) Recursive step: If u e L then aaubeL ii) Closure: A string w is in L only if it can be obtained from the basis by a finite number of applicatious of the recursive step. a) Give the sets La, Li, and L2 generated by the recursive definition. b) Give an implicit definition of the set of strings defined by the recursive definition. c) Prove by mathematical inductioa that for every string in L, the number of a's in is twice the mmber b's in w. Let now)-and nb(1) denote the number of a's and the number of b's inu, respectively. You have a fr Cloud