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Discrete Structures - 7 Questions in Exam | CS 381, Exams of Discrete Structures and Graph Theory

Material Type: Exam; Professor: Toida; 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 Test
July, 2006
1. Fill in the blanks with the SHORTEST string of characters so that the
resultant proposition is valid. [20]
(a) ¬(PQ) ¬(¬PQ)
P ¬Q
(b) (P(PQ)(PT ) (PQ)
P( T Q)
PTP
(c) P(QR) ¬P(QR)
¬P ¬(¬(QR))
¬(QR) ¬P
2. State each of the following formulas in English,if it is a wff. If it is
not a wff, then give a reason why it is not a wff. Here L(x, y) means xis
larger than y,I(x) means xis an integer and the universe is the set of real
numbers: [12]
(a) x¬∃yL(x, y)
For every number, there is no number that is less than that number.
i.e. Every number is not greater than nany number.
(b) xy¬L(x, y)
For every number there is a number that is not less than that.
i.e. Every number is not greater than some number.
(c) xy[[I(x)I(y)] L(x, y)]]
For every number x there is a number y such that if x and y are integer, then
x is gretaer than y.
i.e. For every integer, there is an integer that is less than that.
i.e. Every integer is greater than some integer.
(d) xI(y)L(I(x), y)
This is not a proposition (well-formed formula) because an atomic formula
I(x) is an argument of another atomic formula L(x, y).
pf3

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CS 381 Test

July, 2006

  1. Fill in the blanks with the SHORTEST string of characters so that the resultant proposition is valid. [20] (a) ¬(P → Q) ⇔ ¬(¬P ∨ Q) ⇔ P ∧ ¬Q (b) (P ∨ (P ∧ Q) ⇔ (P ∧ T ) ∨ (P ∧ Q) ⇔ P ∧ ( T ∨ Q) ⇔ P ∧ T^ ⇔^ P (c) P → (Q ∧ R) ⇔ ¬P ∨ (Q ∧ R) ⇔ ¬P ∨ ¬(¬(Q ∧ R)) ⇔ ¬(Q ∧ R) → ¬P
  2. State each of the following formulas in English, if it is a wff. If it is not a wff, then give a reason why it is not a wff. Here L(x, y) means x is larger than y, I(x) means x is an integer and the universe is the set of real numbers: [12] (a) ∀x¬∃yL(x, y) For every number, there is no number that is less than that number. i.e. Every number is not greater than nany number.

(b) ∀x∃y¬L(x, y) For every number there is a number that is not less than that. i.e. Every number is not greater than some number.

(c) ∀x∃y[[I(x) ∧ I(y)] → L(x, y)]] For every number x there is a number y such that if x and y are integer, then x is gretaer than y. i.e. For every integer, there is an integer that is less than that. i.e. Every integer is greater than some integer.

(d) ∀x∃I(y)L(I(x), y) This is not a proposition (well-formed formula) because an atomic formula I(x) is an argument of another atomic formula L(x, y).

  1. Negate the following sentences in English. DO NOT simply say ”It is not the case that ...” or something similarly trivial. [12] (a) Every number is greater than 0. Some number is not greater than 0.

(b) Some number is greater than or equal to every number. Every number is less than some number.

(c) Some numbers are even only if they are integer. Every number is even and not integer.

  1. Express the assertions given below as a proposition of a predicate logic using the following predicates. The universe is the set of numbers.[12] E(x): x is even. I(x): x is integer. (a) Not all numbers are integer.

¬∀xI(x)

(b) A (every) number is even only if it is integer.

∀x[E(x) → I(x)]

(c) It is not necessary for a number to be integer that it is even.

¬∀x[I(x) → E(x)], which I prefer, or ∀x¬[I(x) → E(x)]