Principles of Analysis
Homework Assignment #4
Due Wednesday, February 13, START of Class
Homework should be turned in at the BEGINNING OF CLASS.
All problem numbers refer to Bilodeau and Thie's book, the required text for
the course. You should write up solutions neatly to all problems, paying particular
attention to your arguments and proofs. A nonempty subset will be graded.
You are strongly encouraged to work
on these problems with other classmates, although the solutions you turn in
should be your own work.
Problems on Cardinality and Infinite Sets:
- Explain in your own words Cantor's Diagonalization argument proving that
the set of real numbers is uncountable.
- Show that the union of two countable sets is countable.
- Show that the countable union of countable sets is countable.
In other words, suppose that you have a collection of sets A_n, n belonging to N,
such that A_n is a countable set for each n.
Show that the infinite (but countable) union of all the sets A_n is
Hint: Try to find an explicit method of counting this set which insures that
every element will be counted.
- Using proof by contradiction, show that the set of irrational numbers, Q^C, is
uncountable. Conclude that the set of irrational numbers is much, much "bigger"
than the set of rationals.
Hint: Use the result from problem #2.
- Show that the open intervals (0,1) and (a,b) have the same cardinality by finding a bijection
between them. You may assume that a < b. Your bijection will be a linear function.
- Show that the open interval (0,1) has the same cardinality as R, the set of real numbers,
by finding a bijection between them. Based on the previous problem, conclude that the entire
real number line and any open interval, no matter how small, have the same size!
Hint: Use a function from my favorite subject, trigonometry.
Problems: 2 (parts c, d, f), 7, 10, 13 (part a), 14
Hint: For problem #10, you may use the inequality
| |a| - |b| | <= |a-b| , true for any real numbers a and b.