EECS 501__________________________PROBLEM SET #1__________________________Fall 2001

ASSIGNED: Sept. 07, 2001. Read Munkres handout on mappings and countable vs. uncountable sets.
DUE DATE: Sept. 14, 2001. Read Stark and Woods pp. 1-12. Some of this material will be review.

THIS WEEK: Basics of set theory and countable vs. uncountable sets.
1. Stark and Woods #1.5. Simple problem using sample space.
2. Stark and Woods #1.6 These first 5 problems should go quickly.
3. Stark and Woods #1.9. Applying the axioms of probability.
4. Stark and Woods #1.10 Exclusive "or" and axioms of probability.
5. Stark and Woods #1.11 Exclusive "or" and axioms of probability.
6. Munkres Handout p. 52, #5 (all 10 parts). Countable vs. uncountable sets.
• PROVE your answers by giving 1-1 correspondences with sets of known cardinalities
• (cardinality can be viewed as the number of elements in the set).

7. Let An be the open interval (0,1/n) for n=1,2,3..., so that An={x:0 < x < 1/n}.
Determine the intersection of An over all n > 0. PROVE your answer.
8. The set of algebraic irrational numbers is the set of irrational numbers
that are zeros of any polynomial of any degree with integer coefficients.
• Is the set of algebraic irrational numbers countably infinite or uncountable?
• PROVE your answer by giving a 1-1 correspondence, as in #6.