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Chapter 3: Problem 63

Give an example of a disjunction that is true, even though one of its component statements is false. Then write the negation of the disjunction and explain why the negation is false.

Short Answer

Expert verified
Disjunction: 'It is raining or it is a sunny day' is true although 'it is raining' is false. Negation of the disjunction: 'It is not raining and it is not sunny' is false.

Step by step solution

01

Craft a Disjunction

Create a statement made up of two parts, i.e., a disjunction. For example, take the disjunction 'It is raining or it is a sunny day'.
02

Determine the Truth Value of the Disjunction

In this example, consider a day where it is not raining but it is sunny. Here, the first statement 'It is raining' is false, but the second statement 'It is a sunny day' is true. Hence, the disjunction 'It is raining or it is sunny' is true.
03

Craft and Analyze the Negation of the Disjunction

The negation of the disjunction 'it is raining or it is a sunny day' is 'it is not raining and it is not a sunny day'. Since it is indeed a sunny day, this statement is false. The reason why the negation is false lies in the negation of a disjunction rule in logic, which says that the negation of 'A or B' equals 'not A and not B'

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