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Chapter 3: Problem 72

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

Short Answer

Expert verified
The negation of the statement 'Some \(A\) are \(B\)' is 'No \(A\) are \(B\)'. An example would be negating 'Some students are athletes' resulting in 'No students are athletes.'

Step by step solution

01

Understanding the Statement

When we say 'Some \(A\) are \(B\)', it implies that there exists at least one element in \(A\) that is also in \(B\). In other words, the intersection of \(A\) and \(B\) is not empty.
02

Finding the Negation

The negation of a statement is the inverse of that statement. Thus, the negation of 'Some \(A\) are \(B\)' will be 'No \(A\) are \(B\).' This means the intersection of \(A\) and \(B\) is empty. In other words, there are no elements in \(A\) that are also in \(B\).
03

Example

For instance, if we say 'Some students are athletes,' the negation of this statement would be 'No students are athletes.' which implies that none of the students are athletes.

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Most popular questions from this chapter

Translate each argument into symbolic form. Then determine whether the argument is valid or invalid. You may use a truth table or, if applicable, compare the argument's symbolic form to a standard valid or invalid form. (You can ignore differences in past, present, and future tense.) If I watch Schindler's List and Milk, I am aware of the destructive nature of intolerance. Today I did not watch Schindler's List or I did not watch Milk. \(\therefore\) Today I am not aware of the destructive nature of intolerance.

Use Euler diagrams to determine whether each argument is valid or invalid. All multiples of 6 are multiples of 3 . Eight is not a multiple of 3 . Therefore, 8 is not a multiple of 6 .

Translate each argument into symbolic form. Then determine whether the argument is valid or invalid. You may use a truth table or, if applicable, compare the argument's symbolic form to a standard valid or invalid form. (You can ignore differences in past, present, and future tense.) If The Graduate and Midnight Cowboy are shown, then the performance is sold out. Midnight Cowboy was shown and the performance was not sold out. \(\therefore\) The Graduate was not shown.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I made Euler diagrams for the premises of an argument and one of my possible diagr芒ms did not illustraate the conclusion, so the argument is invalid.

Exercises 59-60 illustrate arguments that have appeared in cartoons. Each argument is restated below the cartoon. Translate the argument into symbolic form and then determine whether it is valid or invalid. If you do not know how to read, you cannot read War and Peace. If you cannot read War and Peace, then Leo Tolstoy will hate you. Therefore, if you do not know how to read, Leo Tolstoy will hate you.
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