MAT511 homework,
due Oct 14, 2009
(1) Let a1 = 2, a2 = 4, and dene ak for all k 3 by an+2 = 5an+1 6an . Prove that
an = 2n for all natural numbers n.
(2) Use the well-ordering principle (every subset of N has a least element) to construct a
proof that for
MAT511 homework,
Let
,
,
due Sept. 30, 2009
be sets.
1. Prove that
if and only if
2. Prove that
.
if and only if
and
.
3. Prove that
. You may use the results above.
4. Prove that
.
5. Give an example where
necessary on
and
. What conditions are
to ensure
MAT511 homework, due Nov 18, 2009
(1) Give a proof using the Pigeonhole Principle: If ve points are
in or on a square of side length 1, then at least two points are
no farther apart than 22 . Start by drawing a picture.
(2) Dene B A to be the set of all f
MAT511 homework, due Dec 9, 2009
(1) Give a careful proof of the proposition:
If A is an innite set, and X is any other set disjoint from A,
then A X is innite.
[You must exhibit a 1-1 correspondence between A X and
a proper subset of A X .]
(2) Give a ca
MAT511 homework, due Nov 11, 2009
(1) For each of these functions f (x) nd the maximum domain of
denition D R. Then restrict the domain to D on which f is
one-one (choose D as large as possible. What is the range R of
the restricted function? Give a formu
MAT511 homework, due Nov 4, 2009
(1) Suppose that A is a nite set with m elements, and B is a nite
set with n elements.
(a) Find the total number of functions from A to B if
m=n
m>n
m<n
(b) Find the number of one-to-one functions from A to B if
m=n
m
MAT511 homework,
due October 7, 2009
Work all four proofs using induction.
(1) Use induction to prove that, for all natural numbers n,
n(n + 1)(2n + 1)
1 + 22 + 32 + + n2 =
.
6
(2) Use induction to prove that, for all natural numbers n,
n(3n 1)
1 + 4 + 7
MAT511 homework,
due Oct 21, 2009
(1) Prove carefully by induction that the binomial coecients
n
k
=
n!
k !(n k )!
satisfy
n+1
k
(2)
(3)
(4)
(5)
(6)
(7)
=
n
k
+
n
k+1
.
(Remember the convention 0! = 1! = 1.
Current (non-vanity) NYS license plates have the
MAT511 homework, due October 28, 2009
R = the real numbers; N = the natural numbers.
(1) For each of (a), (b), (c), (d) below, give a relation R from
A = cfw_5, 6, 7 to B = cfw_3, 4, 5 which ts the description.
(a) R is not a function.
(b) R is a function