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Discrete Mathematics and its Applications

Kenneth H. Rosen

$Math Studyset Vaia Explanations$ Math

7th Edition · 808 pages

ISBN: 9780073383095

Chapter 1: Q46E (page 16)

The truth value of the conjunction of two propositions in fuzzy logic is the minimum of the truth values of the two propositions. What are the truth values of the statements “Fred and John are happy” and “Neither Fred nor John is happy?”

Short Answer

Expert verified

Truth value of “Fred and John are happy” is 0.4 and truth value of “Neither Fred nor John is happy?” is 0.6.

Step by step solution

01

Definition of Fuzzy logic

Fuzzy logic is a mathematical logic which solves problems with an open, imprecise data spectrum.

02

Calculate truth values

The truth value of Fred is happy, is0.8.

Also, the truth value of John is happy, is0.4.

Conjunction of two propositions in fuzzy logic is the minimum of the truth values of the two propositions.

Disjunction of two propositions in fuzzy logic is the maximum of the truth values of the two propositions.

Hence truth value of “Fred and John are happy” is min(0.8,0.4) = 0.4.

The truth value of “Neither Fred nor John is happy” is max(0.2,0.6) = 0.6.

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Most popular questions from this chapter

What is the negation of each of these propositions?

a) Jennifer and Teja are friends.

b) There are $13$ items in a baker’s dozen.

c) Abby sent more than $100$ text messages every day.

d) $121$ is a perfect square.

Let p, q, and r be the propositions

p : You have the flu.

q: You miss the final examination.

r : You pass the course.

Express each of these propositions as an English sentence.

a) $p \to q$

b) $\neg q \leftrightarrow r$

c) $q \to \neg r$

d) $p \lor q \lor r$

e) $(p \to \neg r) \lor (q \to \neg r)$

f ) $(p \land q) \lor (\neg q \land r)$

Construct a truth table for each of these compound propositions.

a) $(p \lor q) \lor r$

b) $(p \lor q) \land r$

c) $(p \land q) \lor r$

d) $(p \land q) \lor r$

e) $(p \lor q) \land \neg r$

f) $(p \land r) \lor \neg r$

Which of these sentences are propositions? What are the truth values of those that are propositions?

Boston is a capital of Massachusetts
Miami is the capital of Florida
$2 + 3 = 5$
$5 + 7 = 10$
$x + 2 = 11$
Answer this question

Show that $\neg (\neg p)$ and pare logically equivalent.

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