Question

Show that the following languages are not regular.

5. Prove that the following languages are not regular: (a) L = {amb’ak: k sn +1}. (b) L = {a” b'ak : kun +1}. (c) L = {a"b'ak: n=1 or 1 # k}.

Parts(b) and (c)

Answer 1

w into xyz such that lylto, lxy15 b) L = { an beak 1 k&ntd} proof using pumping kemma! Let take a string WEL, wo mabaaa Now divède w into xyz such to a baad 4y/2 our ty pumping umma, for all iso, ayZEL Let isl, sy'z & abaaa EL 1-2, ayz - abbana L since we have i=2, where xy²z &2, which violates pomping lomma, Hence is not regular, Now, c) L = fan blak 1 nal or lak} proof using pumping limmar- het take a string wel, we abaa Now, divide w isto xyz, such that lylto (24) £4 Now, by pumping linna for all i>o, ayZ EL Let P-1, xyz abaa EL 1:2, xylz abbaa & Lince, we have in 2, where xy&2, which violates pumping lomma, Hence is not regular Scanned with CamScanner

Umo01 Oxes ! pomping lemma for reg war longeingest Let be a regular lengrage. Then, there exists a constant a(whose value depends upon 2) such that for every string we belenging to the language Lend (whan, w can be broken into 3 parts x,y,z such that wonyz, 476, and 1xylan, Then for all 970, the string xyiz cs aliou belongs to CamScanner