Question

(a) Show that P is closed under union and intersection. That is, show that for all A, B E P AUB,AnBEP (b) Show that NP is closed under union and intersection.

Answer 1

α) class p is Cles "dunder union : LEP L cp be Supposed. M, and MPoly homial- time TMs ト1, Yunning time 6(nM), H2 Yunning time h-tength ot input w; kりkaase Constants Ms-uing Machine åecides LiULz M3on input .uwith nputw. EA M, arrepts, accept , . Run Ma with input ω· 2t M2 accepts, orept ctheulse , veject e) Mg accepts w tf and only if both Mi a1Ha accepts us . )Ovrell Yoming time of M3 is Pokynomial as Sm of tuo toly nomials Polnemial Class P Glosed under in-krsection Here Same but ke Gons truct a dedder M with Polynamial time Complexity deciding Linl: M: an input . Run Mi on input x 2-1f H, accepted, then on M.nǐnputz, dsc reret s If ta olsa accepted Ihen occept, etse reject tn Norst rose M wilt un beth M ana M2, in which Cose it use nt o(m) steps. Let ke mokjki). Thee See has time Compexity O(n) and hence ns LiDL EP.

bClass uP s Closed undey UNION Unien: M on input . Run M on TE occepted then accept 3Ese eject The lenyest bronch, in amy Gemputatian tre on input to art length n i5 on mox(ro. So l ts poly- tine non deteninistic decider foy LiUl2 K don't un , ond M 'n parallel as įtons necessały es: าn.the Picof that recogni7able longuages are Clased undet unien Another po ible Cons thuction would be to non determiniSticaly choose ether M or Ms and Simulate onts Grlected maChine. ドPclassis Closed under INTERSECT! oN : Inkrbrction: H.-on input to : t: Run an TE, rejected then ejrct 2 Else vun Mz n Tt t rejected then reedt. 3 EiSe Accept Cleorly, longest bwanch in Com putation tree on inpt a of de ter mi 5HC decider por Linh.