Discrete Structures
HW #10 —
§§2.3–2.4: Functions, Sequences, & Summations
Last Update: 13 November 2010
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All exercises come from, or are based on exercises from, the Rosen text.
Each HW problem's solution should consist of:
All solutions must be handwritten.
PUT YOUR NAME, DATE, RECITATION SECTION, &
"HW #10"
AT TOP RIGHT OF EACH PAGE;
STAPLE MULTIPLE PAGES 
 (3 points each; total = 9 points)
p. 146: 12 a–c
 You are given some functions and asked to determine which
are
1–1.
 For full credit, you must justify your answers.
 (3 points each; total = 9 points)
pp. 146–147: 14 a, c, e
 You are given some functions and asked to determine which
are
onto.
 For full credit, you must justify your answers.
 (3 points each; total = 6 points)
p. 147: 32
 You are given 2 functions and asked to compute both ways
of composing them with each other.
 For full credit, you must show all your work.
 (6 points)
p. 148: 68.
 You are asked to prove that a certain kind of function is
1–1 iff
it is onto.
 Hint: Use the definitions of "1–1" and of "onto"
together with the fact that:
if A = B, then:
if a function from one of these to the other is not 1–1
or not onto, then A ≠ B.
 (3 points)
p. 149: 74a
 Be sure to read the section on partial functions on
p. 149, column 1, starting immediately after problem 72 and
ending at the end of the column.
 You should probably try to do problems 73a,c–e and check your
answers before attempting problem 72.
 You are asked to prove that a partial function (i.e.,
a function that is not defined on some elements of its
domain) can be "extended" to a total function by assigning
an arbitrary image to each element for which it is
not defined.
 Hint: All you have to do is show that
f^{*} satisfies the definition of a function.
 (3 points each; total = 9 points)
p. 161: 4 a–b, d
 For 3 sequences, you are asked to compute the first 4
terms.
 (3 points)
List the first 10 terms of the sequence {a_{n}}
whose first two terms are a_{0} = –3 and
a_{1} = 2, and which is such that each
succeeding term is the sum of the two previous terms.
 I.e., a_{n+2} =
a_{n} + a_{n+1}
 (3 points)
p. 161: 16a
 You are asked to compute the value of a summation
(a.k.a., a "series"), i.e., to compute the sum of the
terms of a sequence.
 Hint: Compute the terms of the sequence,
and then add them up!
Total points = 48
Tentative grading scheme:
A 4648
A 4445
B+ 4143
B 3840
B 3637
C+ 3335
C 2832
C 2227
D+ 1721
D 916
F 0 8
DUE: AT THE BEGINNING OF LECTURE, FRI., NOV. 19 
Text copyright © 2010 by William J. Rapaport
(rapaport@buffalo.edu)
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http://www.cse.buffalo.edu/~rapaport/191/F10/hw10.html20101113