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# Fuzzy logic is used in artificial intelligence. In fuzzy ISBN: 9780073383095 37

## Solution for problem 46E Chapter 1.1

Discrete Mathematics and Its Applications | 7th Edition

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Problem 46E

Fuzzy logic is used in artificial intelligence. In fuzzy logic a proposition has a truth value that is a number between 0 and 1, inclusive. A proposition with a truth value of 0 is false and one with a truth value of 1 is true. Truth values that are between 0 and 1 indicate varying degrees of truth. For instance, the truth value O.8 can be assigned to the statement “Fred is happy,” because Fred is happy most of the time, and the truth value 0.4 can be assigned to the statement “John is happy,” because John is happy slightly less than half the time. Use these truth values to solve Exercises 45–47.

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?”

Step-by-Step Solution:

Step 1</p>

Given ,The truth value  to the statement “ Fred is happy “  is 0.8

And  the truth value to the statement “ John is happy  is  0.4

Step 2</p>

We have to find the truth values to the statements

“ Fred and John are happy”   and   ‘   Neither Fred nor John is happy “

Step 3</p>

We have the truth value of the conjunction of two propositions in fuzzy logic is the minimum  of  the truth values of  the two propositions.

Step 4</p>

Truth value to the statement

“ Fred and John are happy” is ,

Truth value  to the statement ‘ Fred is happy ‘  is 0.8

Truth value  to the statement ‘ john is happy ‘  is 0.4

Therefore truth value to the statement “ Fred and John are happy” is

Minimum { 0.8 ,0.4 } = 0.4

Step 5 of 7

Step 6 of 7

##### ISBN: 9780073383095

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