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Solutions Exam 1 - Discrete Structures | CS 381, Exams of Discrete Structures and Graph Theory

Old Dominion UniversityDiscrete Structures and Graph Theory

Material Type: Exam; Class: DISCRETE STRUCTURES; Subject: Computer Science; University: Old Dominion University; Term: Unknown 1989;

Typology: Exams

Pre 2010

Uploaded on 02/12/2009

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CS 381 Solutions to Test 1
October 5, 2004
1. Convert the following statements to if then form in English:
(a) I go to the beach only if it is sunny.
If I go to the beach then it is sunny.
(b) For me to go to the beach it must be sunny.
If I go to the beach then it is sunny.
(c) Whenever it is sunny, I go to the beach.
If it is sunny, then I go to the beach.
(d) I go to the beach on a sunny day. [16]
If it is sunny, then I go to the beach.
2. Find the converse and contrapositive of each of the statements of Question
1 above. [24]
Converse:
(a) If it is sunny, then I go to the beach.
(b) If it is sunny, then I go to the beach.
(c) If I go to the beach then it is sunny.
(d) If I go to the beach then it is sunny.
Contrapositive:
(a) If it is not sunny, then I don’t go to the beach.
(b) If it is not sunny, then I don’t go to the beach.
(c) If I don’t go to the beach then it is not sunny.
(d) If I don’t go to the beach then it is not sunny.
3. Negate the following statements in English. Give a form other than simply
putting ”not” or ”it is not the case that” in front:
(a) If it is sunny, I will go to the beach.
It is sunny but (and) I will not go to the beach.
(b) Someone has read every book in the library on this subject.
pf3

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Download Solutions Exam 1 - Discrete Structures | CS 381 and more Exams Discrete Structures and Graph Theory in PDF only on Docsity!

CS 381 Solutions to Test 1

October 5, 2004

  1. Convert the following statements to if then form in English: (a) I go to the beach only if it is sunny. If I go to the beach then it is sunny. (b) For me to go to the beach it must be sunny. If I go to the beach then it is sunny. (c) Whenever it is sunny, I go to the beach. If it is sunny, then I go to the beach. (d) I go to the beach on a sunny day. [16] If it is sunny, then I go to the beach.
  2. Find the converse and contrapositive of each of the statements of Question 1 above. [24] Converse: (a) If it is sunny, then I go to the beach. (b) If it is sunny, then I go to the beach. (c) If I go to the beach then it is sunny. (d) If I go to the beach then it is sunny. Contrapositive: (a) If it is not sunny, then I don’t go to the beach. (b) If it is not sunny, then I don’t go to the beach. (c) If I don’t go to the beach then it is not sunny. (d) If I don’t go to the beach then it is not sunny.
  3. Negate the following statements in English. Give a form other than simply putting ”not” or ”it is not the case that” in front: (a) If it is sunny, I will go to the beach. It is sunny but (and) I will not go to the beach. (b) Someone has read every book in the library on this subject.

Everyone has not read some book in the library on this subject. (c) Either it is raining, or it is snowing and the sun is shining. It is not raining, and either it is not snowing or the sun is not shining. (d) It is raining but the sun is shining. [16] It is not raining or the sun is not shining.

  1. Fill in the blanks with the shortest string of characters so that the resultant proposition is valid. [15] [[P → Q] ∧ [Q → R]] → [P → R] ⇔ ¬[[P → Q] ∧ [Q → R]] ∨ ¬P ∨ R ⇔ ¬[P → Q] ∨ ¬[Q → R] ∨ ¬P ∨ R ⇔ ¬[¬P ∨ Q] ∨ ¬[ ¬Q ∨ R] ∨ ¬P ∨ R ⇔ [P ∧ ¬Q ] ∨ [Q ∧ ¬R] ∨ ¬P ∨ R ⇔ [P ∧ ¬Q] ∨ ¬P ∨ [Q ∧ ¬R] ∨ R ⇔ [[P ∨ ¬P ]∧[¬Q∨¬P ]]∨[[Q∨R]∧[¬R ∨ R]] ⇔ [ T ∧ [¬Q ∨ ¬P ]] ∨ [[Q ∨ R] ∧ T ] ⇔ T

5 (a) Express the argument given below using the symbols suggested for each proposition. [5] (b) Find where the glasses are using inference rules on the propositions in symbolic form. [12] T → B L ∨ K L → C ¬B K → T

Argument: If my glasses are on the kitchen table(T), then I saw them at breakfast(B). I was reading the newspaper in the living room (L)or in the kitchen(K). If I was reading the newspaper in the living room, then my glasses are on the coffee table(C). I did not see my glasses at breakfast. If I was reading the newspaper in the kitchen, then my glasses are on the kitchen table. (a) T → B L ∨ K L → C