Math 455.1 Homework Set 2 Spring, 2009
Problems from textbook quoted in their entirety here
Due: Monday, February 23 (start of class)
ā¢For this homework set, you should again work in a team of 2 or 3 and turn in a single paper
for the entire team. It might be a good idea to form diļ¬erent teams than for Set 1. See
further details on the AboutāHomework Sets page of the course web site.
For Mathematica work here, turn in printed pagesājust whatās directly relevant. You may,
and in fact are encouraged, to place associated written work directly onto such printed pages
(provided itās neat and easy to ļ¬nd).
ā¢See further instructions about homework format on the web site.
1. The text uses algebraic manipulations to arrive at the Addition Formula:
īn
kī=īnā1
kā1ī+īnā1
kī(k=1,2,...,nā1,n=2,3,...)
Give another justiļ¬cation of this formula, by interpreting the three binomial coeļ¬cients in-
volved in terms of number of subsets.
2. Review Exercise 1.8.21 on page 23: Let Bbe a subset of A,|A|=n, |B|=k. What is the
number of all subsets of Awhose intersection with Bhas 1 element?ā
In the three problems below, arrive at the ļ¬nal answer:
(i) ļ¬rst, by using principles taught in this course, not brute-force enumeration of all
possibilities, stating. . .
ā¢which combinatorics principle or principles are being used [for example, #(AĆ
B)=#(A)#(B); the Principle of Inclusion-Exclusion; etc];
ā¢what the relevant mathematical model is in terms of sets or functions (for
example, a product of sets; the set of all functions from one set to another;
etc.); and
ā¢what the constituent sets are that go into the model (for example, which ļ¬nite
sets you are forming the product of; the sets constituting the domain and
codomain for the set of functions). . .
(and you may use Mathematica to do the arithmetic to get an actual numeric
answer); and then, as a check,
(ii) by forming the set of all possibilities through use of relevant functions from note-
books SetsAndFunctions.nb or Combinatorics.nbādo not print the lists con-
stituting these sets, please!āand then using Card to count these possibilities.
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