University of Illinois Spring 2009
ECE 313: Problem Set 2
Sets, Events, Axioms of Probability and Their Consequences
Due: Wednesday February 4 at 4 p.m..
Reading: Ross Chapter 1, Sections 1-4; Chapter 2, Sections 1-5
Powerpoint Lecture Slides, Sets 1-6
Noncredit Exercises: Chapter 1: Problems 1-5, 7, 9;
Theoretical Exercises 4, 8, 13; Self-Test Problems 1-15.
Chapter 2: Problems 3, 4, 9, 10, 11-14;
Theoretical Exercises 1-3, 6, 7, 10, 11, 12, 16, 19, 20; Self-Test Problems 1-8
Yes, the reading and noncredit exercises are the same as in Problem Set 1.
1. [Two useful MacLaurin series]
In Problem 1(d),(e) of Problem Set 1, you showed (we hope!) that (1+x)n=
n
X
k=0
n(n−1) · · · (n−k+ 1)
k!xk
for positive integer n. Now, according to Eq. 4.2 of Ross Chapter 1 (with y= 1),
(1 + x)n=
n
X
k=0 n
kxkwhere n
k=n!
k!(n−k)!.
(a) Show that your answer to Problem 1 of Problem Set 1 agrees with the result in the textbook.
(b) Now consider the function g(x) = (1 −x)−nwhere nis a positive integer. Does the MacLaurin
series for g(x) contain terms of degree > n? If not, what is the highest degree term?
(c) Use the result of part (b) to write down the MacLaurin series for (1 −x)−1and (1 −x)−2. These
results will be used so often in ECE 313 that we recommend that you memorize them.
2. [Subsets of a finite set]
Let Ω denote a finite set containing the nelements ω1, ω2, . . . , ωn. The cardinality (more informally,
the size) of a subset A⊂Ω is the number of elements in A, and is denoted as |A|.
(a) Let n= 4. List al l the subsets of Ω in increasing order of size. How many subsets are there? How
many of these subsets are non-empty subsets?
(b) If you listed only 14 or 15 subsets in part (a), please re-do part (a), and this time, include the
empty set ∅and/or Ω as subsets of Ω.
(c) In your answer to part (a) or (b), verify that for each k, 0 ≤k≤4, the total number of subsets of
size kis the same as the total number of subsets of size 4 −k. Now explain why for nin general,
the total number of subsets of size kis the same as the total number of sets of size n−k.
(d) Each subset Acorresponds to a n-bit vector (x1, x2, . . . , xn) where xi= 1 if ωi∈Aand xi= 0
if ωi/∈A. Writing A↔(x1, x2, . . . , xn) emphasizes the one-to-one correspondence: each subset
defines a unique n-bit vector, and each n-bit vector defines a unique subset, e.g. with n= 4, we
have that {ω1, ω3} ↔ (1,0,1,0).
i. What n-bit vectors correspond to Ω and to ∅? What n-bit vector corresponds to Ac?
ii. If A↔(x1, x2, . . . , xn), B↔(y1, y2, . . . , yn), (A∪B)↔(z1, z2, . . . , zn)
and (A∩B)↔(w1, w2, . . . , wn), express the zi’s and wi’s in terms of the xi’s and yi’s. Hint:
you may need the logical operators ∨and ∧that you may have encountered in ECE 290.
iii. How many different n-bit vectors are there? How many different subsets are there of Ω?
iv. “They correspond to the nonempty subsets of Ω” Respond as if you are on JeopardyTM: What
is the question to which this statement is the answer? How many subsets of Ω are non-empty?
