Question

Let A= = {* : nen}u(2,3). PA Please mark all true statements (there might be more than one statement that is true LEO A for all nEN. The interior of Ais Int(A)={0}U(2,3). Points and 2 are accumulation points of A. The boundary of Ais A = {0,2,3}. The closure of A in R is A=Au{0,2,3).

Answer 1

ht) n n+1 = 2n(n+1) Now, n-l n+! -P, ( ) ЛА so, The statement : The interior of A Solution :- A= }ř:n nem ,U (2,3) (i) ÞEA. Take P = 1/2 < ħ - Patik htP<h heat-pt +) so, of A =) ht => h E OA [ AS DA= Ã - int(A)] Thus, This id Int(A)= 10) u (2,3) is FALSE ñ + p) But, -p, +D), A ñ is not an interior point of a - int(A) [ As, Ā = A UA' Where A be the derived set of A ] ,مر :: EOA, KNEIN • This option option is True hi claim, o & Int (A) Take, t= Ź, Then (0-Ź, 0+ 2) = (-tit) is an open neighbourhood of o. Now, ft ( -², ² ) but - 13 & A. so (-ta, ²) & A => 0 & int a

claim, 0 is a accumulation ecumulation point. het, €70. Then by archimedian Broferty 7 KGN such that te LE 'SO, *tt-t,t) => for any tyo IKEA such that kt (0-t, ott) 160-4,044) - +, 0++) – 103n A + P => 0 is a accumulation point of A. claim, 2 is not a accumulation foint Let, Exo Then 27€ > 2 so, (2, 2+) (2,3) + => (2-4, 2+t) n A ¢ => 12-t, 2+t) NA -12) +9 Thus, 0, 2 are accumulation point of A. Joo, The statement; "Points 0 and 2 are accumulation points at Anis True . id Now, ŕ EOA , HNEN . But, ñ ¢ 30,2,3] => JA & 30,2,37 Bo, The Stalment jp DA= 10,2,31" is FALSE b of A The bandary

Take, to be any real number. (V) Ā = Now, where a' is the derived set of A 0,2 are limit of A. point of A. Claim, 3 is a limit point of a 3 Ther, (3-6, 3 ç (2, 3) => (2,3) n (3-€, 3+€) ° 2, 3) n (3-6,3+t)}- 33374 => (2,3) => 3 is a limit point of t. 0,2,3 EA' => Ā Z AUAI => 2 AU 10,2,37 so, The closure of a ink is Ā = AU 30,2,3} statement is True. correes aption E TEOA, for au nem so, The area Thus, The points of A. are accumulation A point o and 2 The elesure of A in Ris Ā= AU30,2,3}