What is the negation of the statement

yxgao

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!!

Doc Al

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.

Hurkyl

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]

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yxgao	What is the negation of the statement 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!!

Hurkyl Recognitions: Gold Member Science Advisor Staff Emeritus	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]