Math 310: Homework 7 (Ch 14) – due Wednesday, 11/26
In pbls 3 and 4 below, you may assume that the Continuum Hypothesis holds.
1. Use Cantor’s diagonal argument to write a complete formal proof showing that
the interval of real numbers [2,3] is uncountable.
2. Prove that if Aand Bare infinite countable sets, then A∪Bis also countable.
You may assume Aand Bare disjoint.
3. Determine the cardinality of the following sets. Your answer should be either an
integer number, or one of ℵ0,ℵ1,ℵ2, etc. No proof is needed (but you should be
able to justify your answer if asked!).
a) the irrational numbers
b) Q×Q
c) S={√n|n∈Q≥}
d) T={m
√n|n∈N, m ∈N}
e) A={n∈Z|0≤n≤41}
f) the power set of the rational numbers, ℘(Q)
g) the complex numbers: C={x+iy |x, y ∈R, i =√−1}
4. Give examples of sets (other than precisely N81,N,Z,Qor R) with cardinality:
a) 81
b) 0
c) ℵ1
d) ℵ5
1