1549901369-Elements_of_Real_Analysis__Denlinger_

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Answers & Hints for Selected Exercises 705


  1. (a) n(-n,n)=(-1,1) LJ (-n,n) = (-oo,+oo)
    nEN nEN
    u (-n, n)c = (-oo, -1] u [1, +oo) n (-n, n)c = 0
    nEN nEN
    (b) n ( -oo, n) = ( -oo, 1) U (-oo, n) = (-oo, +oo)
    nEN nEN
    LJ (-oo, n)c = [1, +oo) n (-oo,n)c =^0
    nEN nEN
    (c) n (-, ) = {O}
    nEN


u (-*'*) = (-1,1)
nEN
c
U (-*, *t = (-oo, 0) U (0, +oo)
nEN

c
n (-*, *t = (-oo, -1] u [1, +oo)
nEN

(d) n [-2+*,2-*J=[-1,1J
nEN

u [-2 + * ,2-*] = (-2,2)
nEN
LJ [-2+*,2-*t=(-oo,-l)U(l,oo)
nEN
n [-2+ *,2-*t = (-oo,-2] u [2,oo)
nEN
(e) n (n , n + 1) = 0 U (n,n+l)=(l,+oo)-N
nEN nEN
n (n, n + l)c = (-oo, -1] UN
nEN


  1. XE ( LJ A,)c 9 X rf_ LJ A.x
    .-\EA ,\EA
    9 ,..., :L\ E A 3 x E A>. 9 '<! >. E A , x rf-A>.
    9 '<!>.EA , x EA~ 9 x E n A~.
    .-\EA

  2. x EB n ( LJ A>.) 9 x EB and x E LJ A>.
    .-\EA .-\EA
    9 x E B and 3>. E A 3 x E A>. 9 3>. E A 3 x E B and x E A>.
    9 3>. EA 3 x EB n A,\ 9 x E u (B n A>.)·
    .-\EA


EXERCISE SET B.2


  1. (a) V (f) =IR; R(f) =IR; 1-1; onto
    (b) V (f) =IR; R(f) = [-2, oo); not 1-1; not onto
    (c) V (f) = [~, oo); R(f) = [O, oo); 1-1; not onto
    ( d) V (f) =IR; R(f) = [O, oo ); not 1 -1; not onto
    (e) V (f) =IR - {O}; R(f) = {-1, 1}; not 1-1; not onto
    (f) V (f) =IR; R(f) = (0, 1); not 1-1; not onto

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