Due Sept 12: Sect 1.1, Nos. 1,2,6,7.

Due Sept 19: Sect 1.2, Nos. 1,2,3,6,8

Sect 1.3, Nos. 1,2,3,4,7 Note misprints: 2(c) should read:
"The union of * two* countable sets is countable." 7 should read:
"Suppose that for each * natural number * n the set..." 8(c)
should read: "The set of functions from a countable set to a
*finite* set."

Due Sept 26: Sect 2.1, Nos. 1,2,5,6

Sect 2.2, Nos. 2,4,5,8

Sect 2.4, Nos. 1,6,7

*and:*

The *Catalan numbers* are the sequence defined by
C(1) = 1, C(2) = 1, and the recursion relation:

C(n) = C(1)C(n-1) + C(2)C(n-2) + ... + C(n-2)C(2) + C(n-1)C(1).
So for example C(3) = 2, C(4) = 5, C(5) = 14, etc.

Let 2n points be placed around the circumference of a circle
and labelled 0,1,2,...,2n-1. An *n-chord configuration*
is a set of n non-intersecting chords in the circle, with
end points at 0,1,2,...,2n-1. Prove that for any integer n,
the number N(n) of distinct n-chord configurations is equal to
the Catalan number C(n+1).

*Hint: Let P(n+1)=N(n)
to get the indices to match; set P(1) = N(0) = 1; show that
P(2) = 1; and show that the P's satisfy the same recursion
relation as the C's. To do this count separately the configurations
in which the chord from 0 goes to 1, to 3, to 5, etc. Finally
use induction to show that the same start and the same recursion
relation implies that the two sequences are the same.*

Due Oct 3: Sect 2.5, Nos. 1(a,b),3,4

Sect 2.6, Nos. 1,3,4.

No homework for October 10 (Midterm Exam).

Due October 17: Sect 3.1, Nos. 1,2,6,8,9,11,12.

Due October 24: Sect 3.2, Nos. 1,2,3,5,6 and the following
exercises in ``logic"

a) give the negation of the statement: ``Every person in
this room has at least ten dollars."

b) give the negation of the statement: ``Every person in this
room who has at least ten dollars is at most eighteen years
old."

c) for each of the two definitions we have of ``f is continuous
at x" state the negation: what it means for f NOT to be
continuous at x.

d) take the definition in the book of ``f is uniformly
continuous on the interval (a,b)" and state its negation:
what it means for f to be NOT uniformly continuous.

e) take the definition of ``L is the limit of the sequence
a_n" (a sub n) and state its negation: how do you know when
L is NOT the limit of the sequence a_n?

Due October 31: Sect 3.2, Nos. 7,8,9,10,11.

Due November 7: Sect 3.3, Nos. 1,2,3,4,7,10

Sect 3.4, No. 1.

Due November 14: Sect 3.5, Nos. 1,2,3,5ab,6,7a,9.

Due November 21: Sect 4.1, Nos. 1,4,5,6,8

Sect 4.2, Nos. 1,3,5,6,7,8,11

Due *Tuesday* December 10: Sect 4.3, Nos. 1,2 (interpret ``the
first few" as ``the first five"),3,5,6 (use polynomials at *x*_0 = 9)

Sect 4.4 Nos. 1,2 (you may use your calculator for a rough location
of the roots, and for implementing the steps of Newton's method), 6.

Due December 12: Sect 5.1, Nos. 1,2,3,8,12

Sect 5.2, Nos. 2,4,5,6

Sect 9.2, Nos. 1,2,3.