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Chapter 3: Problem 71

Explain how to write the negation of a quantified statement in the form "All \(A\) are \(B\)." Give an example.

Short Answer

Expert verified
The negation of a quantified statement 'All A are B' is 'There exists an A that is not B.' Example: Negation of the statement 'All dogs are animals' is 'There exists a dog that is not an animal.'

Step by step solution

01

Understanding Quantified Statements

In mathematics, a quantified statement is one that makes a claim about a certain amount or all of a specific set. Here, 鈥淎ll A are B鈥� is a universal quantified statement, indicating that every member of set A is also a member of set B.
02

Negating the Universal Quantified Statement

The negation of a universally quantified statement 鈥淎ll A are B鈥� will assert that there is at least one member of A that is not B. In other words, it's saying not every member of set A is in set B. So the negation of the statement is 'There exists an \(A\) such that it is not \(B\).'
03

Giving an Example

To further illustrate, consider a statement like 鈥淎ll dogs are animals.鈥� The negation of this statement would be 鈥淭here exists a dog that is not an animal.鈥� This means that there is at least one 'dog' that does not belong to the category of 'animals.'

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