134 9 Number Systemsabove by bl//'n >- 1.
c) If t >• 1 and n >- f^j, then 6^1 /" x i.
d) If w 3 6"' X y, then bw+l/n -< y for sufficiently large n.
e) If fe"J x ?y, then fr'""^1 /" >_. ^ for sufficiently large n.
f) Let 71 = {u; e K : 61U X t/}. Show that x = sup/1 satisfies 6* = y.
g) Prove that x above is unique.
9.5. Let F be an ordered field. Prove the following:
a) x, y & F and x^2 + y^2 = 0 =^- x = 0 and y — 0.
b) xi, x'2, • • •, .x'n G F and .xf + • • • + x^ = 0 => xi = x-i = • • • = x„ = 0.
9.6. Let m be a fixed integer. For a, b 6 Z, define a ~ 6 if a — 6 is divisible by
m, i.e. there is an integer fc such that a — b = mk.
a) Show that ~ is an equivalence relation in Z.
b) Describe the equivalence classes and state the number of distinct equiva-
lence classes.
9.7. Do the following:
a) Let X = R, and x ~ y if x £ [0,1] and y £ [0,1]. Show that ~ is symmetric
and transitive, but not reflexive.
b) Let. X / I and ~ is a relation in X. The following seems to be a proof
of the statement that if this relation is symmetric and transitive, then it is
necessarily reflexive:
x ~ y => y ~ x, x ~ y and y ~ x ==> x ~ x;therefore, x ~ x, Vx £ X. In view of part a), this cannot be a valid proof.
What is the flaw in the reasoning?
9.8. Prove the following:
a) If Xy, X2, • • •, Xn are countable sets, then X — II]l=:1Xi is also countable.
b) Every countable set is numerically equivalent to a proper subset of itself.
c) Let X and Y be non-empty sets and / : X 1—> Y be an onto function. Prove
that if X is countable then Y is at most countable.
Web material
http://129.118.33.l/~pearce/courses/5364/notes_2003-03-31.pdf
http://alpha.fdu.edu/~mayans/core/real_numbers.html
http://comet.lehman.cuny.edu/keenl/realnosnotes.pdf
http://en.wikipedia.org/wiki/Complex_number
http://en.wikipedia.org/wiki/Countable
http://en.wikipedia.org/wiki/Field_(mathematics)
http://en.wikipedia.org/wiki/Numeral_system
http://en.wikipedia.org/wiki/Real_number