HW #10 — §§2.3–2.4: Functions, Sequences, & Summations

All solutions must be handwritten.

1. (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.

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2. (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.

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3. (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.

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4. (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|.

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5. (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.

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6. (3 points each; total = 9 points)

   p. 161: 4 a–b, d

   • For 3 sequences, you are asked to compute the first 4 terms.

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7. (3 points)

   List the first 10 terms of the sequence {a_n} whose first two terms are a₀ = –3 and a₁ = 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

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8. (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       46-48
A-      44-45
B+      41-43
B       38-40
B-      36-37
C+      33-35
C       28-32
C-      22-27
D+      17-21
D        9-16
F        0- 8