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.
- Stark and Woods #1.5. Simple problem using sample space.
- Stark and Woods #1.6 These first 5 problems should go quickly.
- Stark and Woods #1.9. Applying the axioms of probability.
- Stark and Woods #1.10 Exclusive "or" and axioms of probability.
- Stark and Woods #1.11 Exclusive "or" and axioms of probability.

- 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).

- Let A
_{n} be the open interval (0,1/n) for n=1,2,3..., so that A_{n}={x:0 < x < 1/n}.

Determine the intersection of A_{n} over all n > 0. PROVE your answer.

- 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.