Math 455.1 Homework Set 1 Spring, 2009
Due: Friday, February 13 (start of class)
•For this homework set, you should work in a team of 2 or 3 and turn in a single paper for the
entire team. See further details on the About→Homework Sets page of the course web site.
For Mathematica work here, turn in printed pages—just what’s directly relevant. You may,
and in fact are encouraged, to place associated written work directly onto such printed pages
(provided it’s neat and easy to find).
•See further instructions about homework format on the web site.
1. (a) Using a relevant formula, and without actually enumerating the subsets involved, calcu-
late:
i. The number of 1-element subsets of a 10-element set.
ii. The number of 2-element subsets of a 10-element set.
iii. The number of 4-element subsets of a 10-element set.
iv. The number of 6-element subsets of a 10-element set.
v. The number of 8-element subsets of a 10-element set.
vi. The number of 10-element subsets of a 10-element set.
vii. The number of subsets of a 10-element set.
(b) Now, for k= 0,1,2,...,10:
•exhibit all the k-element subsets of a 10-element set;
•actually count the number of such k-element subsets (you may use Card); and
•verify that this number is 10
k.
Use relevant functions described in SetsAndFunctions.nb or Combinatorics.nb to form
the sets of these subsets.
2. Do Review Exercise 1.8.17, page 22 of the textbook.
3. Do the following exercise from notebook SetsAndFunctions.nb: In terms of functions already
discussed above, define a function SymmetricDifference that finds the symmetric difference
of two sets. Your definition should take the form SymmetricDifference[x , y ] := . . . , and
the result of SymmetricDifference[setA,setB] for two sets setA and setB should be their
symmetric difference.
4. (a) For subsets Aand Bof a set X, express the characteristic function cA△Bof the symmetric
difference of Aand Bin Xin terms of the characteristic functions cAand cBof Aand
B, respectively, in X.
(b) Prove that symmetric difference is associative, that is, for all subsets A, B, C of a set X,
we have A△(B△C) = (A△B)△C. Use the result of (a) along with the fact that two
subsets of Xare equal when their characteristic functions are the same.
5. Use relevant functions described in SetsAndFunctions.nb or Combinatorics.nb to form the
sets of injections described. In (c), use Card to count the number of injections.
(a) Exhibit all injections from {1,2}to {1,2,3,4,5}.
(b) Exhibit all injections from {1,2,3,4,5,6}to {1,2,3,4,5}. Account for the result.
(c) For k= 1,2,3,4,5,6, calculate the number of injections from a k-element set to a 5-
element set and verify that this number is P(5, k).