Chapter 3.4, Problem 76E

In symbolic logic, a statement is either tumor false (consider true to have value of 1 and false a value of 0). In fuzzy logic, nothing is true a false, but everything is a matter of degree. For example, consider the statement “The sun is shining.” In fuzzy logic, this statement may have a value between 0 and 1 and may be constantly changing. For example, if the sun is partially blocked by clouds, the value of this statement may be 0.25. In fuzzy logic, the values of connective statements are round as follows for statements p and q.

Not p has a truth value of 1 − p.

p ˄ q has a truth value equal to the minimum of p and q.

p ˅ q has a truth value equal to the maximum of p and q.

p → q has a truth value equal to the minimum of 1 and 1 − p + q.

p ↔ q has a truth value equal to 1 − qp − qq, that is. 1 minus the absolute value of p minus q.

Absolute values are discussed in Section 12.6

Suppose the statement “p: The sun is shining” has a truth value of 0.25 and the statement “q: Mary is getting a tan” has a truth value of 0.20. Find the truth value of

a. ~p

b. p˄ q

c. ~q

d. ~ q

e. p ˅ q

f. p ↔ q