Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 268 — #276
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Chapter 8
Infinite Sets
(b) Show that for each type of path, either
(i) the f -arrows define a bijection between the A and B elements on the path, or
(ii) the g-arrows define a bijection between B and A elements on the path, or
(iii) both sets of arrows define bijections.
For which kinds of paths do both sets of arrows define bijections?
(c) Explain how to piece these bijections together to form a bijection between A
and B.
(d) Justify the assumption that A and B are disjoint.
Problem 8.11. (a) Prove that if a nonempty set C is countable, then there is a total
surjective function f W N ! C .
(b) Conversely, suppose that N surj D, that is, there is a not necessarily total
surjective function f W ND. Prove that D is countable.
Problem 8.12. (a) For each of the following sets, indicate whether it is finite,
countably infinite, or uncountable.
(i) The set of even integers greater than 10 100 .
(ii) The set of “pure” complex numbers of the form ri for nonzero real numbers r.
(iii) The powerset of the integer interval Œ10::10 10 .
(iv) The complex numbers c such that 9m; n 2 Z: .m C nc/c D 0.
Let U be an uncountable set, C be a countably infinite subset of U, and D be
a countably infinite set.
(v) U [ D.
(vi) U \ C
(vii) U
D
(b) Given examples of sets A and B such that
R strict A strict B:
Recall that A strict B means that A is not “as big as” B.