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Smith et al - Advanced Math - 8/e (Homework)

James Finch

Math - College, section 1, Fall 2019

Instructor: Dr. Friendly

Current Score : 31 / 54

Due : Sunday, January 27, 2030 00:00 EST

Last Saved : n/a Saving...  ()

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Question
Points
1 2 3 4 5 6 7 8 9
21/21 2/4 1/4 4/4 3/4 –/12 –/3 –/1 –/1
Total
31/54 (57.4%)

  • Instructions

    The best-selling A Transition to Advanced Mathematics, 8th edition, published by Cengage Learning, helps students bridge the gap between calculus and advanced math courses.

    Please read the information in the box at the top of each exercise to learn more about the features, content, and/or grading. We encourage you to try correct and incorrect answers in any answer blanks. This demo assignment allows many submissions and allows you to try another version of the same question for practice wherever the problem has randomized values.

    WebAssign provides a wide range of exercises that enable you to:
    • Develop Conceptual Understanding (#1-3: Read Its, practice fundamentals of mathematical logic)
    • Build Proof-Writing skills (#4-6: engage in automatically-graded proof exercises)
    • Grow Problem Solving Skills (#7-9: practice advanced mathematical thinking)

Assignment Submission

For this assignment, you submit answers by question parts. The number of submissions remaining for each question part only changes if you submit or change the answer.

Assignment Scoring

Your last submission is used for your score.

1. 21/21 points  |  Previous Answers SmithAdvMath8 1.1.010c. My Notes
Question Part
Points
Submissions Used
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
1/1 1/1 1/1 1/1 1/1 1/1 1/1 1/1 1/1 1/1 1/1 1/1 1/1 1/1 1/1 1/1 1/1 1/1 1/1 1/1 1/1
1/100 1/100 1/100 1/100 1/100 1/100 1/100 1/100 1/100 1/100 1/100 1/100 1/100 1/100 1/100 1/100 1/100 1/100 1/100 1/100 1/100
Total
21/21
 
  • This exercise will build conceptual understanding.
  • Students practice with the fundamentals of mathematical logic to build their conceptual understanding.
  • Read It links are available as a learning tool under each question so students can quickly jump to the corresponding section of the eTextbook.
Use a truth table to determine whether the following is a tautology, a contradiction, or neither.
(P ∧ Q) ∨ (~P ∨ ~Q)
P Q ~P ~Q P ∧ Q ~P ∨ ~Q
(P ∧ Q) ∨ (~P ∨ ~Q)
T T Correct: Your answer is correct. Correct: Your answer is correct. Correct: Your answer is correct. Correct: Your answer is correct. Correct: Your answer is correct.
F T Correct: Your answer is correct. Correct: Your answer is correct. Correct: Your answer is correct. Correct: Your answer is correct. Correct: Your answer is correct.
T F Correct: Your answer is correct. Correct: Your answer is correct. Correct: Your answer is correct. Correct: Your answer is correct. Correct: Your answer is correct.
F F Correct: Your answer is correct. Correct: Your answer is correct. Correct: Your answer is correct. Correct: Your answer is correct. Correct: Your answer is correct.
     Correct: Your answer is correct.
You're right!
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2. 2/4 points  |  Previous Answers SmithAdvMath8 1.2.014. My Notes
Question Part
Points
Submissions Used
1 2 3 4
1/1 0/1 1/1 0/1
2/100 1/100 1/100 1/100
Total
2/4
 
  • This exercise will build conceptual understanding.
  • Students practice with the fundamentals of mathematical logic to build their conceptual understanding.
(a)
Give, if possible, an example of a false conditional sentence for which the converse is true.
     Correct: Your answer is correct.
Outstanding!
(b)
Give, if possible, an example of a false conditional sentence for which the converse is false.
    
(c)
Give, if possible, an example of a false conditional sentence for which the contrapositive is false.
     Correct: Your answer is correct.
Perfect!
(d)
Give, if possible, an example of a false conditional sentence for which the contrapositive is true.
     Incorrect: Your answer is incorrect.
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3. 1/4 points  |  Previous Answers SmithAdvMath8 1.3.006. My Notes
Question Part
Points
Submissions Used
1 2 3 4
0/1 1/1 0/1 –/1
2/100 1/100 1/100 0/100
Total
1/4
 
  • This exercise will build conceptual understanding.
  • Students practice with the fundamentals of mathematical logic to build their conceptual understanding.
Let
T = {17}, U = {6}, V = {24}, and W = {2, 3, 7, 26}.
 In which of these four different universes is the statement true? (Select all that apply.)
(a)
(∃x) (x is odd right double arrow implies x > 8).
Incorrect: Your answer is incorrect.

(b)
(∃x) (x is odd ∧ x > 8).
Correct: Your answer is correct.

That's it!
(c)
(∀x) (x is odd right double arrow implies x > 8).
Incorrect: Your answer is incorrect.

(d)
(∀x) (x is odd ∧ x > 8).

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4. 4/4 points  |  Previous Answers SmithAdvMath8 1.4.005g. My Notes
Question Part
Points
Submissions Used
1 2 3 4
1/1 1/1 1/1 1/1
1/100 2/100 1/100 1/100
Total
4/4
 
  • This exercise will build skill in proof writing.
  • A variety of automatically-graded exercise types engage students in completing proofs by different methods.
Let x and y be integers. Prove the following.
If x and y are odd, then xy is odd.
Construct a proof for the statement by selecting sentences from the following scrambled list and putting them in the correct order.
  • Since 2𝘬𝘮 is an integer, 𝘹𝘺 is odd.
  • Then there are integers 𝘬 and 𝘮 such that 𝘹 = 2𝘬 and 𝘺 = 2𝘮.
  • Since 2𝘬𝘮 + 𝘬 + 𝘮 is an integer, 𝘹𝘺 is odd.
  • Then there are integers 𝘬 and 𝘮 such that 𝘹 = 2𝘬 + 1 and 𝘺 = 2𝘮 + 1.
  • Then 𝘹𝘺 = (2𝘬)(2𝘮) = 2(2𝘬𝘮).
  • Then 𝘹𝘺 = (2𝘬 + 1)(2𝘮 + 1) = 2(2𝘬𝘮 + 𝘬 + 𝘮) + 1.
  • Suppose that 𝘹 and 𝘺 are odd integers.
Proof.
1. Correct: Your answer is correct. 2. Correct: Your answer is correct. 3. Correct: Your answer is correct. 4. Correct: Your answer is correct.
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5. 3/4 points  |  Previous Answers SmithAdvMath8 1.5.001f. My Notes
Question Part
Points
Submissions Used
1 2 3 4
1/1 1/1 0/1 1/1
2/100 1/100 1/100 2/100
Total
3/4
 
  • This exercise will build skill in proof writing.
  • A variety of automatically-graded exercise types engage students in completing proofs by different methods.
Analyze the logical form of each of the following statements, and construct just the outline of a proof by the given method. Do not provide any details of the proof.
Outline a two-part proof that a subset A of the real numbers is compact if and only if A is closed and bounded.
Proof.
    • Suppose A is compact
    • ⋯
    • Therefore, Correct: Your answer is correct. .
    • Suppose Correct: Your answer is correct. .
    • ⋯
    • Therefore, Incorrect: Your answer is incorrect. .
We conclude that A is compact if and only if Correct: Your answer is correct. .
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6. –/12 points SmithAdvMath8 1.5.003g. My Notes
Question Part
Points
Submissions Used
1 2 3 4 5 6 7 8 9 10 11 12
–/1 –/1 –/1 –/1 –/1 –/1 –/1 –/1 –/1 –/1 –/1 –/1
0/100 0/100 0/100 0/100 0/100 0/100 0/100 0/100 0/100 0/100 0/100 0/100
Total
–/12
 
  • This exercise will build skill in proof writing.
  • A variety of automatically-graded exercise types engage students in completing proofs by different methods.
Let x be an integer. Write a proof by contraposition to show the following.
If 8 does not divide
x2 − 1,
 then x is even.
Proof. Suppose x is . Then
x =
for some integer
m.
 Then
x2 − 1 = 
2
 
 − 1 = = .
 Since if
xy
 is odd, then both
x
 and
y
 are odd, is , so = for some integer
k.
 Thus
x2 − 1 = 4
 = ,
 so 8 divide
x2 − 1.
 Therefore, if 8 does not divide
x2 − 1,
 then
x
 is even.
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7. –/3 points SmithAdvMath8 3.1.010abd. My Notes
Question Part
Points
Submissions Used
1 2 3
–/1 –/1 –/1
0/100 0/100 0/100
Total
–/3
 
  • This exercise will grow problem solving skills.
  • Many exercises seek to have students discover their own examples to show mathematical traits and relationships.
Let
A = {a, b, c, d}.
 Give an example of relations R and S on A such that we have the following. (Enter the relations R and S in set notation as a comma-separated list. Enter each relationship
x R y
 as
(x, y)).
(a)
R ∘ S ≠ S ∘ R
R, S =
(b)
(S ∘ R)−1 ≠ S−1 ∘ R−1
R, S =
(c)
R and S are nonempty, and R ∘ S and S ∘ R are empty.
R, S =
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8. –/1 points SmithAdvMath8 3.4.002d. My Notes
Question Part
Points
Submissions Used
1
–/1
0/100
Total
–/1
 
  • This exercise will grow problem solving skills.
  • Students practice their advancing mathematical thinking across a variety of more advanced mathematical topics that are introduced through the course.
Perform the calculation in ℤ9.
3 · 5 + 4 · 6
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9. –/1 points SmithAdvMath8 4.5.007b. My Notes
Question Part
Points
Submissions Used
1
–/1
0/100
Total
–/1
 
  • This exercise will grow problem solving skills.
  • Students practice their advancing mathematical thinking across a variety of more advanced mathematical topics that are introduced through the course.
Let
f: ℝ → ℝ
 be given by
f(x) = 
2x        if x ≥ 1
2 − 2xif x < 1
.
subsets A and C of ℝ such that
f(A ∩ C) ≠ f(A) ∩ f(C)
Give an example of the above. (Enter A and C as single intervals.)
A, C =
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Enter an exact number.