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MATH 1019 Discrete Math Midterm Exam 1 Review Questions with Verified Solutions, Exams of Discrete Mathematics

York CollegeDiscrete Mathematics

CS/MATH 1019 Discrete Math for Computer Science MATH 1019 Midterm Exam Review Questions with Verified Solutions | 100% Pass | Graded A+ |

Typology: Exams

2024/2025

Available from 10/25/2024

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SC/MATH 1019B
Solutions to Test 3
Nov. 29th 2024
Page 1 of 8
Name:
YorkU email:
Student Number:
READ THE FOLLOWING INSTRUCTIONS.
•
Do not open your exam until told to do so.
•No calculators, cell phones or any other electronic devices can be used on this exam.
•
Clear your desk of everything excepts pens, pencils and erasers.
•
If you need scratch paper, the last page is blank.
Show all your work unless otherwise indicated. Write your answers clearly!
Include enough steps for the grader to be able to follow your work.
•
You will be given exactly 60 minutes for this exam.
I have read and understand the above instructions:
SIGNATURE
•
pf3
pf4
pf5
pf8

Partial preview of the text

Download MATH 1019 Discrete Math Midterm Exam 1 Review Questions with Verified Solutions and more Exams Discrete Mathematics in PDF only on Docsity!

Name : YorkU email : Student Number : READ THE FOLLOWING INSTRUCTIONS.

- Do not open your exam until told to do so.

  • No calculators, cell phones or any other electronic devices can be used on this exam.
  • Clear your desk of everything excepts pens, pencils and erasers.
  • If you need scratch paper, the last page is blank. Show all your work unless otherwise indicated. Write your answers clearly! Include enough steps for the grader to be able to follow your work.
  • You will be given exactly 60 minutes for this exam. I have read and understand the above instructions: SIGNATURE

Extra Work Space. Multiple Choice. Circle the best answer. No work needed. No partial credit available.

  1. (5 points) Which of are the following relations on { 1 , 2 , 3 } is symmetric? A. {(1 , 2) , (2 , 3) , (2 , 1) , (3 , 2)} B. {(1 , 1) , (2 , 2) , (3 , 3) , (1 , 3)} C. {(1 , 2) , (2 , 3) , (3 , 1)}
  2. (5 points) The graph G = ( V, E ) where E = {{ 1 , 2 } , { 2 , 3 } , { 3 , 4 } , { 1 , 4 } , { 2 , 4 }} and V = { 1 , 2 , 3 , 4 } is: A. a bipartite graph B. NOT a bipartite graph
  3. (5 points) If an is the number of bit strings which do not contain four consecutive 0’s, then an satisfies which recurrence: A. an = an − 1 + 2 an − 2 + 3 an − 3 B. an = an − 1 + an − 2 + an − 3 + an − 4 C. an = 2 an − 1 + 2 n −^1 D. an = 3 an − 2 + an − 4.

Solution: We can define the set A by the following properties:

- λ ∈ _A

  • w_ 1 ∈ A if w ∈ _A
  • w_ 011 ∈ A if w ∈ A The bit strings of length 5 in A are: 01111 , 10111 , 11011 , 11111 Standard Response Questions. Show all work to receive credit.
  1. (15 points) Consider the set Σ∗^ of bit strings (i.e. Σ = 0 , 1 ). Let A Σ∗^ be the set of bit strings such that any 0 is followed by at least two 1’s. Give a recursive definition of the set A and list all bit strings of length 5 that are in A.

2 2 Solution: The characteristic equation is r 2 −r += (r −)(r − 3)

and so bn = A

n 2

  • B(3)n for some constants A and B. The initial conditions give us A + B = 2 1 A + 3B = 3 2 which implies −5B = −4. So, we see that B = 4/5 and A = 6/5. Therefore bn =

n +(3)n.

  1. (20 points) Solve the recurrence relation bn = 7 bn − 1 − 3 bn − 2 with b 0 = 2 and b 1 = 3.

Solution: The graph W 7 looks like 1 7 2 8 6 3 5 4 on the left with vertices labeled and on the right with a proper coloring with 4 colors. Since the vertices {1, 2, 3, 4, 5, 6, 7 } make an odd length cycle these vertices need at least 3 colors. Since the vertex 8 is connected to all other vertices it needs to be its own color. Therefore we can conclude the chromatic number of W 7 is 4

  1. (20 points) Let Wn be the graph with vertex set V = { 1 , 2 ,... , n + 1 } and edge set E = {{ i, n + 1 } : 1 ≤ i ≤ n } ∪ {{ i, i + 1 } : 1 ≤ i ≤ n − 1 } ∪ {{ 1 , n }}. (a) Draw the graph W 7. (b) Find the chromatic number of W 7.

Extra Work Space.