Question

(The First Partition Theorem). For any ACR, we have: (16.20) Theorem 16.1.11 Int(A) U a(A) U Ext(A) = R; Int(A) n a(A) Ø; Int(A)n Ext(A) = Ø; a(A)n Ext(A) = Ø. Proof. The proof simply boils down to writing the definitions of the sets in the right way and applying De Morgan's Laws for each of the three steps. (xER E> 0 s.t. N(x, E)n (R \ A) = Ø}; Int(A) (16.21) (xERE> 0 s.t. N(x, e)n A = Ø}; a(A)={xER Ie> o, Nx , e )n (R \ A) Ø Ext(A) and N(%, ε) nΑ 4 6) .

Please follow the recommendations suggested and use de Morgan's laws. I would like know how de Morgan's law is used to proof the theorem.

Answer 1

EntA)U aU Ent{k) Thun bly tet Oy either ERA then eithur Tuka) oy Thin Thun eyTE Ext(A) YRA SCRNA) Thevefore in ot Case EEnt ZO)0 Eutla) From DAU EtA R\nk ExtA) TR\ Cnt A U Ent(AU aA) RA Ex HA)

RaDeMngan's (aw A UERLA U .EntA) IntA)nExtA C A UFtA U In t(A) J'd-A) =((omu(Gru))) h tA) UEt (A) U Ite) UdA) A n ERA (Extra) C TRC-