Docsity
Log inSign up
Docsity
Prepare for your exams
Prepare for your exams

Study with the several resources on Docsity

Earn points to download
Earn points to download

Earn points by helping other students or get them with a premium plan

Guidelines and tips
Guidelines and tips
Sell on Docsity
Sell on Docsity
Log inSign up

Prepare for your exams

Study with the several resources on Docsity

Find documents

Prepare for your exams with the study notes shared by other students like you on Docsity

Search Store documents

The best documents sold by students who completed their studies

Search through all study resources
Docsity AINEW

Summarize your documents, ask them questions, convert them into quizzes and concept maps

Explore questions

Clear up your doubts by reading the answers to questions asked by your fellow students

Earn points to download

Earn points by helping other students or get them with a premium plan

Share documents
download point icon20 Points

For each uploaded document

Answer questions
download point icon5 Points

For each given answer (max 1 per day)

All the ways to get free points
Get points immediately

Choose a premium plan with all the points you need

Study Opportunities

Choose your next study program

Get in touch with the best universities in the world. Search through thousands of universities and official partners

Community

Ask the community

Ask the community for help and clear up your study doubts

University Rankings

Discover the best universities in your country according to Docsity users

Free resources

Our save-the-student-ebooks!

Download our free guides on studying techniques, anxiety management strategies, and thesis advice from Docsity tutors

From our blog

Exams and Study
Go to the blog

MATH 1019 Discrete Math Exam Test 2 Review Questions with Verified Solutions, Exams of Discrete Mathematics

York CollegeDiscrete Mathematics

CS/MATH 1019 Discrete Math for Computer Science

Typology: Exams

2024/2025

Available from 10/25/2024

star_score_grades
star_score_grades 🇺🇸

3.6

(19)

1.7K documents

1 / 10

Toggle sidebar

This page cannot be seen from the preview

Don't miss anything!

bg1
SC/MATH 1019B
Solutions to Test 1B
Oct. 2nd 2024
Page 1 of 10
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.
•Without fully opening the exam, check that you have pages 1 through 10.
•
Fill in your name, etc. on this first page.
Show all your work unless otherwise indicated. Write your answers clearly!
Include enough steps for the grader to be able to follow your work.
Do first all of the problems you know how to do immediately. Do not spend too
much time on any particular problem. Return to difficult problems later.
•
You will be given exactly 60 minutes for this exam.
I have read and understand the above instructions:
SIGNATURE
•
•
pf3
pf4
pf5
pf8
pf9
pfa

Partial preview of the text

Download MATH 1019 Discrete Math Exam Test 2 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.
  • Without fully opening the exam, check that you have pages 1 through 10.
  • Fill in your name, etc. on this first page. Show all your work unless otherwise indicated. Write your answers clearly! Include enough steps for the grader to be able to follow your work. Do first all of the problems you know how to do immediately. Do not spend too much time on any particular problem. Return to difficult problems later.
  • 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) The compound proposition ( p ∨ q ) ∧ ( p → (¬ q )) is: A. Satisfiable. B. Unsatisfiable.
  2. (5 points) The negation of ∃ x ∃ y ( x + y = 7) is: A. ∀ x ∃ y ( x + 7 = 0) B. ∃ x ∃ y ( x + y /= 7) C. ∀ x ∀ y ( x + y /= 7)
  3. (5 points) A proposition logically equivalent to p → q is: A. ¬ p → ¬ q B. q → p C. ¬ q → ¬ p

Solution: Standard Response Questions. Show all work to receive credit.

  1. (10 points) Construct a truth table for the compound proposition (¬ p ) → ( p ∨ q ∨ r ). p q r ¬ p p ∨ q ∨ r (¬ p ) → ( p ∨ q ∨ r ) T T T F T T T T F F T T T F T F T T T F F F T T F T T T T T F T F T T T F F T T T T F F F T F F

Solution: ¬( p ∨ r ∨ (¬ p ∧ q )) ≡ ¬ p ∧ ¬ r ∧ ¬(¬ p ∧ q ) ≡ ¬ p ∧ ¬ r ∧ ( p ∨ ¬ q ) ≡ (¬ p ∧ ¬ r ∧ p ) ∨ (¬ p ∧ ¬ r ∧ ¬ q ) ≡ F ∨ (¬ p ∧ ¬ r ∧ ¬ q ) ≡ ¬ p ∧ ¬ r ∧ ¬ q ≡ ¬ p ∧ ¬ q ∧ ¬ r

  1. (15 points) Show using a chain of logical equivalences that ¬( p ∨ r ∨(¬ p ∧ q )) is equivalent to ¬ p ∧¬ q ∧¬ r.

Solution: One possible solution is S = {x, y, z, w} and T = {1, 2, 3, 4 }. The fact that |P(S)| = 16 means |S| = 4 and |S × T | = 16 means |S||T | = 16. In this case the members of S × T are:

  1. (10 points) Provide examples of two finite sets S and T so that |P( S )| = 16 and | S × T | = 16. List all members of set S × T for your chosen sets S and T. ( x, 1) ( y, 1) ( z, 1) ( w, 1) ( x, 2) ( y, 2) ( z, 2) ( w, 2) ( x, 3) ( y, 3) ( z, 3) ( w, 3) ( x, 4) ( y, 4) ( z, 4) ( w, 4)

Solution: We first see when x = 0, for any y ∈ Q we have that xy = 0 · y = 0 /= 1. Thus we can Now take any x ∈ Q with x = 0. This means x = conclude that ∃x, ∀y(xy /= 1) is true. We must now show the uniqueness. definition of a rational number b = 0, and a a b = 0 because x for a, b ∈ Z with both a /= 0 and b /= 0 (by the = 0). For such x we can then take y = b a which is also a rational number (hence in our domain) and see that xy = 1. Therefore we can conclude that ∃!x, ∀y(xy /= 1) is true as desired.

  1. (15 points) Let x and y be from the domain consisting of all rational numbers. Prove that the statement ∃! x, ∀ y ( xy /= 1) is true.

Extra Work Space.