Math 215: Homework for week 13, with new problems 44a,44b Prof. S. Smith due Monday 15 April 2002

Replace problem 44 (on the original assignment) with the following two problems, based on 2 and 4 of Exam 2 given Fri 5 April:

Problem 44a: Decide if the following functions are injective and/or surjective—by giving a proof: (i) f : R^{2} → R^{2} defined by f(x_{1}, x_{2}) = (x_{2}, 2x_{2}). (ii) f : R^{2} → R^{2} defined by f(x_{1}, x_{2}) = (x_{1} + x_{2}, x_{1} − x_{2}).

Problem 44b: Show that the following sets are denumerable (countably infinite), by giving a bijection with (possibly a subset of) Z^{m} or N^{m} for a suitable m > 1:

(i) the set of “Pythagorean triples”: that is, all (a, b, c) with a, b, c ∈ N such that a^{2} + b^{2} = c^{2}. (One example is the triple 3, 4, 5).

(ii) the set of integer 2 × 2 symmetric matrices: that is, all ( \begin{matrix}a & b \\ b & c \end{matrix} ) with a, b, c ∈ Z. (One

example is the matrix ( \begin{matrix}5 & 8 \\ 8 & −12 \end{matrix} )

Problem 45: (essentially 12.1 in the text)

Write down the first 10 rows of Pascal’s triangle. use this to find the coefficient of a^{7}b^{3} in (a + b)^{10}.

Problem 46: (Similar to 12.2 in the text)

Find the coefficient of x^{4} in (x + 2)^{7}.

Problem 47: (12.4 in the text)

Three people each select a main dish from a menu of 5 items. How many choices are possible (i) if we record who selected which dish (as the waiter should) (ii) if we ignore who selected which dish (as the chef could) ?

Problem 48: (12.6 in the text)

Prove that the product of any n consecutive positive integers is divisible by n!.

Problem 49: (p. 185, no. 20)

Use the pigeonhole principle to prove that, given a set S of 10 distinct positive integers less than 107, there exist two disjoint subsets of S with the same sum.