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Examination 1 - Discrete Mathematics | MAT 340, Exams of Discrete Mathematics

University of Southern Mississippi (USM)Discrete Mathematics
Prof. Samuel Lyle

Material Type: Exam; Professor: Lyle; Class: Discrete Mathematics; Subject: Mathematics; University: University of Southern Mississippi; Term: Unknown 2010;

Typology: Exams

2009/2010

Uploaded on 02/25/2010

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Unit 1 Test
MAT 340
Name:
Directions: Please include all work done in the process of getting a solution. No notes or other materials are
allowed. Extra paper is available upon request.
Problem Possible Points Points Earned
1 10
2 10
3 10
4 10
5 12
6 16
7 10
8 10
9 12
Total 100
Problem 1 (10 points):
Fill in the remaining entries in the following truth table
p q r (p∧q) (p∧q)∨r ((p∧q)∨r)→p
TTT TT T
T T F T T T
T F T FT T
F T T F TF
T F F F F T
F T F F F T
F F T F T F
F F F F F T
1
pf3
pf4

Partial preview of the text

Download Examination 1 - Discrete Mathematics | MAT 340 and more Exams Discrete Mathematics in PDF only on Docsity!

Unit 1 Test

MAT 340

Name:

Directions: Please include all work done in the process of getting a solution. No notes or other materials are allowed. Extra paper is available upon request.

Problem Possible Points Points Earned 1 10 2 10 3 10 4 10 5 12 6 16 7 10 8 10 9 12 Total 100

Problem 1 (10 points):

Fill in the remaining entries in the following truth table

p q r (p∧q) (p∧q)∨r ((p∧q)∨r)→p T T T T T T T T F T T T T F T F T T F T T F T F T F F F F T F T F F F T F F T F T F F F F F F T

Consider the statement “If the class is quiet, they will get a longer recess”

  • Write the contrapositive of this statement If the class does not get a longer recess, then they were not quiet
  • Write the converse of this statement If the class got a longer recess, then they were quiet

Problem 3 (10 points):

  • Give an example of a tautology (should be in logical form) and justify it with a truth table.
    • The statement p∨(∼p) is a tautology

p (∼p) p∨(∼p) T F T F T T

  • Give an example of a contradiction (should be in logical form) and justify it with a truth table.
    • The statement p∧(∼p) is a contradiction

p (∼p) p∧(∼p) T F F F T F

Problem 4 (10 points):

  1. Rewrite the following statement as a conditional statement. “Either Pete does not eat lunch, or he eats out.” “If Pete eats lunch, then he eats out” also, “If Pete does not eat out, then he does not eat lunch”
  2. Write out the logical equivalence that this shows. (That is write (p→q) ≡ to an expression involving only ∧,∨, or ∼)

(p→q) ≡ (∼p)∨q

  1. Show that the following two statements are logically equivalent, using logical identities (p∧(p→q)) ≡ (q∧p)

(a) (p∧(p→q)) (b) ≡ (p∧((∼p) ∨q)) (c) ≡ (p∧(∼p)) ∨(p∨q) (d) ≡ (c) ∨(p ∨q) (e) ≡ p ∨q

Write the negations of the two statements, using De Morgan’s law.

  1. Either I will walk to the game, or I will drive. The negation of this statement is “I will not walk to the game, and I will not drive.”
  2. Jim is good at basketball and soccer. The negation of this statement is “Jim is not good at basketball or he is not good at soccer.”

Problem 8 (10 points):

Again, let L(x) be the predicate “x lives in Louisiana”, F(x,y) be the predicate “x is friends with y”, and let D be the set of all students at USM. Determine the negation of the following statements (in symbolic form), and then translate it into English.

  1. ∃ x∈D, such that L(x) The negation is ∀ x∈D, ∼L(x) In English, “Every student at USM is not from Louisiana”
  2. ∀ x∈D, L(x) The negation is ∃ x∈D, ∼L(x) In English, “There is some student at USM who is not from Louisiana”

Problem 9 (12 points):

Write the following statements with any needed universal and existential quantifiers. Separately, state any domains, and the predicate for each case

  1. Given any real number x, there is always another number y so that x · y = 2
    • Domain(s) Real numbers R for both domains
    • Predicate x · y = 2
    • Statement ∀ x∈R, ∃ y∈R such that x · y = 2
  2. Everybody loves somebody
    • Domain(s) P: the set of all people for both domains
    • Predicate x loves y
    • Statement ∀ x∈P, ∃ y∈P such that x loves y
  3. Somebody loves everybody
    • Domain(s) P: the set of all people for both domains
    • Predicate x loves y
    • Statement ∃ x∈P, ∀ y∈P such that x loves y