Question

(3 pts each) For each of the following find an indexed collection {An}nen of distinct sets (no two sets are equal) such that (a) n=1 An = {0} (b) Un=1 An = [0, 1] (c) n=1 An = {-1,0,1} (5 pts each) Give example of an explicit function f in each of the following category with properly written domain D and range R such that (a) There exists a subset S of D with f-'[F(S)] + S (b) There exists a subset T of R with f[f-1(T)] + T (11) (3 + 3 + 5 + 5 + 2) Define functional completeness. Show that x + y = (x+y)+(x+y), x y = (x + 2) + (y + y), ū = (x + x) Is {4} functionally complete? Justify your answer.

Answer 1

ã An= (hehh for NEM. To show An = {04. n21 Ở 4, 430. n=1 nzi nal Lot If possible let Then I ne o Ansuch that ao case 1: Let axo. Then I a MEN such that to n. This af (otho). Then nefroito) Hance, there exists a not in such that ñ An. xed And Then nd herefore Case 2! 23 - Pin Let no. to take So pro crists such that <P. Then, -they-p and so, thon Hence xf (- o) and no, x4 (-ho to? There exists a noen such that af A Therefore Then there ano fa no . nel Hence ato => na n&o Ano a & An anno Since out for any nem and oxyn for any nell, ofthitag onean, OG D An. Hence noi

( 5 ) Now for ܢ [osta] CAI An = [0, 6] A1 = [0, 1). amy nem hele and so, Thus, Anç Ai anaon, Hence = Aulu And FA, = [0, 1]. This part is similar to. is similar to. peart @ Anfin. 8 An na

A m2 @ Let Analhhh ) for nary. Then a Anasoy n=1 6 Let An= [auth] for nein. Then ů An = [0,1]. na © Let An = (x-ths 1 thn) ul-betulleita) 1 n=1 Let sa Theni n An {-19, 0,1}. Let D ={1,2,34, B={2,65. Define ti DR by f(1) = f(2)=a +13) =6. {lg. Then fis)= a. And fo{fess] - {1,2} & S. Let D={1, 2} , Rasab, e Define f:D → R by f(s = a, f(27 = a Let To {a,b]. Then. f (T) = {1,2%. And ff ()] = $$1,24 = {a} # T.
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