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Chapter 1: Q60E (page 36)

Show that the negation of an unsatisfiable compound proposition is a tautology and the negation of a compound proposition that is a tautology is unsatisfiable.

Short Answer

Expert verified

An unsatisfiable compound proposition is always false. The negation of an unsatisfiable compound proposition is then always true. This indicates that the negation of an unsatisfiable compound proposition is a tautology. The tautology is equivalent with Tand its negation is equivalent with F. This shows that the negation of tautology is unsatisfiable.

Step by step solution

01

Definition of the tautology  

A tautology is a compound statement that is true for all feasible truth values of the statements.

02

To show the negation of an unsatisfiable compound proposition is a tautology and the negation of a compound proposition that is a tautology is unsatisfiable.

A composite statement that is unsatisfiable is always false. The negation of an unsatisfiable compound statement is always true in this case. This means that the negation of an unsatisfactory compound statement is a tautology. The tautology is identical to T, and its negation is equivalent to F. This demonstrates that the negation of tautology is unsatisfiable.

Therefore, it has been shown that the negation of an unsatisfiable compound proposition is a tautology and the negation of a compound proposition that is a tautology is unsatisfiable.

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Most popular questions from this chapter

A says “We are both knaves” and B says nothing. Exercises 24–31 relate to inhabitants of an island on which there are three kinds of people: knights who always tell the truth, knaves who always lie, and spies (called normals by Sullying [Sm78]) who can either lie or tell the truth. You encounter three people, A, B, and C. You know one of these people is a knight, one is a knave, and one is a spy. Each of the three people knows the type of person each of other two is. For each of these situations, if possible, determine whether there is a unique solution and determine who the knave, knight, and spy are. When there is no unique solution, list all possible solutions or state that there are no solutions

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