Negating "For All" Statement: Proving False

  • Thread starter keemosabi
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In summary, the conversation is discussing how to negate a statement and prove it false. The statement in question involves multiple conditions and uses "for all" and "there exists" statements. The suggested approach is to change all "for all" statements to "there exists" and to modify the direction of the inequality to "\le". This is based on the rules of negation and the specific conditions in the given statement.
  • #1
keemosabi
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Homework Statement


Negate the following statement and thereby prove that it is false.
[PLAIN]http://img834.imageshack.us/img834/5020/74520479.png

Homework Equations


The Attempt at a Solution


I know the general rules but there are so many conditions in this statement that I don't know how to apply them.

My guess right now would be to change all the "for all" statements to "there existst" and to reverse the direction of the last inequality.
 
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  • #2
keemosabi said:

Homework Statement


Negate the following statement and thereby prove that it is false.
[PLAIN]http://img834.imageshack.us/img834/5020/74520479.png


Homework Equations





The Attempt at a Solution


I know the general rules but there are so many conditions in this statement that I don't know how to apply them.

My guess right now would be to change all the "for all" statements to "there existst" and to reverse the direction of the last inequality.
Not exactly "reverse it". "<" should become "[itex]\le[/itex]". Also that "there exist [itex]\delta> 0[/itex]" should become "for all [itex]\delta> 0[/itex]".
 
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  • #3
HallsofIvy said:
Not exactly "reverse it". "<" should become "[itex]\le[/itex]". Also that "there exist [itex]\delta> 0[/itex]" should become "for all [itex]\delta> 0[/itex]".
Thank you for the reply. I'm just wondering how you came up with that. Could you elaborate a little please?
 

FAQ: Negating "For All" Statement: Proving False

What does it mean to negate a "for all" statement?

Negating a "for all" statement means to express its opposite, that is, to show that there exists at least one counterexample where the statement is false.

What is the process for negating a "for all" statement?

The process for negating a "for all" statement involves changing the universal quantifier "for all" to an existential quantifier "there exists", and negating the statement that follows it.

How do you prove a negated "for all" statement to be false?

To prove a negated "for all" statement to be false, you need to provide a specific example or counterexample that shows the statement to be untrue.

Can a negated "for all" statement ever be true?

No, a negated "for all" statement can never be true. By definition, a negated "for all" statement states that there exists at least one case where the statement is false, which means it cannot be true for all cases.

What is the importance of negating "for all" statements in scientific research?

Negating "for all" statements is important in scientific research because it allows for the identification of exceptions or limitations to a general rule or hypothesis. This helps to refine and improve scientific theories and models, leading to a better understanding of the natural world.

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