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Hunter - Essentials of Discrete Mathematics 2/e (Homework)

James Finch

Math - College, section 1, Fall 2019

Instructor: Dr. Friendly

Current Score : 14 / 16

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

Last Saved : n/a Saving...  ()

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14/16 (87.5%)

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    Here are some textbook questions from Essentials of Discrete Mathematics 2/e by David Hunter published by Jones & Bartlett Learning. Click here for a list of all of the questions coded in WebAssign. 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.

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1. –/1 points HunterDM2 2.2.002. My Notes
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Draw a Venn diagram to show the region A ∩ B'. This region is also denoted A \ B, and is called the set difference, for obvious reasons. (Select all that apply.)
A
B U

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2. –/1 points HunterDM2 2.2.014. My Notes
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How many integers in the set {n is in Z | 1 ≤ n ≤ 300} are divisible by 2, 3, or 5?
integers
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3. –/1 points HunterDM2 2.2.015. My Notes
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Write down all elements of ({1, 2, 3} ∩ {2, 3, 4, 5}) ∪ {6, 7}. (Enter your answers as a comma-separated list.)
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An integer solution to the equation
3x + 4 = 7y
 is an ordered pair of integers (x, y) that satisfies the equation. For example, (1, 1) is one such solution. Write the set of all integer solutions to the equation
3x + 4 = 7y
 in set builder notation.
    
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5. –/2 points HunterDM2 2.2.025. My Notes
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Let E be the set of all even integers, and let O be the set of all odd integers.
(a) Explain why E ∪ O ⊆ Z.
    

(b) Explain why Z ⊆ E ∪ O.
    
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6. –/3 points HunterDM2 3.1.010. My Notes
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Every year, Alice gets a raise of $4,000 plus 3% of her previous year's salary. Her starting salary is $30,000. Give a recurrence relation for S(n), Alice's salary after n years, for n ≥ 0.
S(n) =  
  if n = 0
  + · S(n − 1)    if n > 0
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7. –/7 points HunterDM2 3.4.015. My Notes
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–/7
 
The Lucas numbers L(n) have almost the same definition as the Fibonacci numbers:
L(n) =   
  1if n = 1
  3if n = 2
  L(n − 1) + L(n − 2)  if n > 2.
Let
α = 
1 + 
5
2
 and
β = 
1 − 
5
2
,
 as in Theorem 3.6. Prove that L(n) = α n + β n for all n is in N. Use strong induction.

Proof. First, note that
L(1) = 1 = α + β,
and
α2 + β2 = (α + 1) + 
β +
 = α + β + 2
 =  = L(2).
Suppose as inductive hypothesis that L(i) = α i + β i for all i < k, for some k > 2. Then
L(k)
 		=
L(k − 1) + L
k −
 
    
     =
α k − 1 + β k − 1
 +
    
     =
α k − 2(α + 1) + β k − 2
β +
    
     =
α k − 2(α2) + β k − 2
    
     =
α k +
,
as required.
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