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Discrete Mathematics and its Applications
Found in: Page 35
Discrete Mathematics and its Applications

Discrete Mathematics and its Applications

Book edition 7th
Author(s) Kenneth H. Rosen
Pages 808 pages
ISBN 9780073383095

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Short Answer

Find the dual of each of these compound propositions.

a) p∨¬q

b) p∧(q∨(r∧T))

c) (p∧¬q)∨(q∧F)

a) p∧¬q

b) p∨(q∧(r∨F))

c) (p∨¬q)∧(q∨T)

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :
TABLE OF CONTENTS

Step1:Definition of dual

The dual of a proposition which contains only the logical operators ∨,∧,¬, is the compound proposition obtained by replacing each ∧ by ∨ , each ∨ by ∧ , each T by F, and each F by T.

Step 2: Dual of a)

Dual of p∨¬q is p∧¬q

Step 2: Dual of b)

Dual of p∧(q∨(r∧T)) is .p∨(q∧(r∨F))

Step 3: Dual of c)

Dual of (p∧¬q)∨(q∧F) is (p∨¬q)∧(q∨T)(p∨¬q)∧(q∨T)

Most popular questions for Math Textbooks

Let p and q be the propositions “Swimming at the New Jersey shore is allowed” and “Sharks have been spotted near the shore,” respectively. Express each of these compound propositions as an English sentence

a) ¬q

b) p∧q

c) ¬p∨q

d) p→¬q

e) ¬q→p

f ) ¬p→¬q

g) p↔¬q

h)¬p∧(p∨¬q)

Let p, q, and r be the propositions

p : Grizzly bears have been seen in the area.

q : Hiking is safe on the trail.

r : Berries are ripe along the trail.

Write these propositions using p, q, and r and logical connectives (including negations).

a)Berries are ripe along the trail, but grizzly bears have not been seen in the area. b) Grizzly bears have not been seen in the area and hiking on the trail is safe, but berries are ripe along the trail. c) If berries are ripe along the trail, hiking is safe if and only if grizzly bears have not been seen in the area. d) It is not safe to hike on the trail, but grizzly bears have not been seen in the area and the berries along the trail are ripe. e) For hiking on the trail to be safe, it is necessary but not sufficient that berries not be ripe along the trail and for grizzly bears not to have been seen in the area. f ) Hiking is not safe on the trail whenever grizzly bears have been seen in the area and berries are ripe along the trail.

Find a compound proposition involving the propositional variables \(p,q\) and \(r\) that is true when \(p\) and \(q\) are true and \(r\) is false, but is false otherwise. (Hint: Use a conjunction of each propositional variable or its negation.)

Find the bitwise OR, bitwise AND, and bitwise XOR of each of these pairs of bit strings.

a) 1011110, 0100001b) 11110000, 10101010c) 0001110001, 1001001000d) 1111111111, 0000000000

Show that ¬¬p and p are logically equivalent.
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