Question

Q11. [10, optional] The reverse of a string can be defined recursively as: AR = a, (wa) R = awR for all a e, wel*. Prove that (uv) R = vRuR, for all u, v est, by Mathematical Induction.

Answer 1

To proof it by mathematical induction, we have to show that the condition holds for p(0),taking p(n) is true, we have to proof that p(n+1) is also true..

here is the solution...

let, Size of a soe. lul=0 let, u=€ [Epsilon] CE (UV)* = (EVJR - VR - VER = VRUR um let, Size of Vire.lv1 = 0 let, v=€ & Cur= (he) = UR = ERUR = VEUR Therefore, the condition holds for 4,0 € I* €

is true. а also. 2 R. (aukar let, for (ulang the condition holds.ie. (UV) VER Then, we have to proof that for lul=ntl. the conditim holds let, (w/= n+1 and wall lulen we have to proof, (WV) R = VR WR Now, wys = caur) )]} ) bron uvka [rar=a = UruRa [oo we took, (uvje VER SVR UROR - VROUR [ we tox, VAR. Guyana -VROUR [proved for ful=nti] let, for lvl = n. the condition holds. i.e. (Lv) & VER Now, we have to proof that, for = +1 the let, (w) = nti q w=va, Ivian we have to proof is true is true. NOU (uw) R WRUR P-TO

Cu w) R = cu vage R son is true] to Es true] ntl also, = a (urg [*: fway!.am] aurar [ we [" We took (uue ver a RVR UR [ar-a] (vajtur [r we took voor cure WR GR [proved for 141=nti] is the condition also holds for 2211 lul, lv The condition holds for 14), lupo 27 Taking that the condition holds for Jul, lvlan, we ta have proved that the condition also holds for 141,1v1 = nt The Statement ) fuje - veue ic forum proved

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