What is the negation of the statement

  • Thread starter yxgao
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In summary: So for the statement above, the negation would be "There exists an s in R such that for each r in R, f(r) >0 and g(s) <0."In summary, the negation of the statement "For each s in R, there exists an r in R such that if f(r) >0, then g(s) >0" is "There exists an s in R such that for each r in R, f(r) >0 and g(s) <0." The general method to find the negation of any logical statement is to distribute the negation using the laws of Boolean logic.
  • #1
yxgao
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What is the negation of the statement "For each s in R, there exists an r in R such that if f(r) >0, then g(s) >0."

The answer is "There exists an s in R such that for each r in R, f(r) >0 and g(s) <0."

What is the general method to find the negation of any logical statement?

Thanks!
 
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  • #2
Originally posted by yxgao
What is the general method to find the negation of any logical statement?
While I can't give you a general method, you may find it useful to review the concept of contradictory statements from Boolean logic:

All S is P is contradictory to Some S is not P

No S is P is contradictory to Some S is P

A statement and its contradictory cannot both be true (or both be false). Thus if "All S is P" is not true, then "Some S is not P" must be true. Of course, this only applies to statements that can be put in standard categorical form.
 
  • #3
Basically, you just want to distribute the negation. Use the laws

[tex]\neg \forall x: P(x) = \exists x: \neg P(x)[/tex]
[tex]\neg \exists x: P(x) = \forall x: \neg P(x)[/tex]
[tex]\neg(x \wedge y) = \neg x \vee \neg y[/tex]
[tex]\neg(x \vee y) = \neg x \wedge \neg y[/tex]
[tex]\neg(x \Rightarrow y) = x \wedge \neg y[/tex]
[tex]\neg(\neg x) = x[/tex]
 
Last edited:

What is the negation of the statement?

The negation of a statement is the opposite or the denial of that statement. It is a way of expressing the opposite meaning of a statement.

Why is the negation of a statement important in science?

In science, the negation of a statement is important because it allows us to consider alternative hypotheses and possibilities. It helps us to think critically and analyze different perspectives, leading to a better understanding of the subject matter.

How is the negation of a statement written?

The negation of a statement can be written in different ways, depending on the context. In logic, it is often denoted by the symbol "~" or "¬". In plain language, it can be expressed using words such as "not", "opposite", or "contradictory to".

Can the negation of a statement be true?

Yes, the negation of a statement can be true. For example, if the statement "All birds can fly" is true, then the negation of this statement "Not all birds can fly" is also true. In science, we often use the negation of a statement to test the validity of a hypothesis.

How can I apply the concept of negation in my scientific research?

In scientific research, you can apply the concept of negation by considering alternative explanations or hypotheses for your findings. This will help you to evaluate and analyze your results more critically and arrive at more reliable conclusions.

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