Is there a y that does not exist for all x such that y^2 = x?

  • Thread starter 1MileCrash
  • Start date
In summary, the conversation discusses how to create a useful negation for the statement "for all x > 0, there exists y > 0 s.t. y^2 = x". Suggestions are made and discussed, with the conclusion that a counter example would need to exist for the statement to be negated.
  • #1
1MileCrash
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Homework Statement



Give a useful negation for:

[itex]\forall x > 0, \exists y > 0 s.t. y^2 = x[/itex]

Homework Equations





The Attempt at a Solution



I'm not sure how to do this, I have

[itex]\exists y > 0 s.t. \forall x > 0, y^2 = x[/itex]

Which says "there exists a y that for all x, y^2 = x"...which is obviously incorrect, as there is no y that when squared equals EVERYTHING.
 
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  • #2
How about, "there exists an x greater than zero such that for all y greater than zero, y squared is not equal to x"
 
  • #3
If you were looking for an example to counter this (a counter example), what would need to be true for that example ? ... and, wouldn't it take just one example (at least as regards x)?
 
  • #4
SammyS said:
If you were looking for an example to counter this (a counter example), what would need to be true for that example ? ... and, wouldn't it take just one example (at least as regards x)?

What do you mean counter it? I can't counter the first statement, it's true.
 
  • #5
SammyS said:
If you were looking for an example to counter this (a counter example), what would need to be true for that example ? ... and, wouldn't it take just one example (at least as regards x)?

This supports my suggestion.
 
  • #6
A negation of "for all x, this is true" is "there exists an x such that this is not true." You say "there exists an x such that for all (for any arbitrary) y this is not true" because it has to be not true for all y. if it's only not true for some y, then there exists a y such that it is true, and you've lost it.
 
  • #7
1MileCrash said:
What do you mean counter it? I can't counter the first statement, it's true.
Well yes, it is true for x & y being real numbers. That doesn't mean that you can't coming up with criteria that would need to hold for a counter-example, if such existed.
 
  • #8
Okay, I get it! So what you're saying, is that if there WERE a counter example, it would satisfy Arcana's negation statement, correct?
 
  • #9
That's what I'm saying if indeed, Arcana's negation statement is correct.
 

1. How can we help negate a statement?

To help negate a statement, you can provide evidence or counterarguments that contradict the original statement. You can also ask for clarification or further information to challenge the validity of the statement.

2. What does it mean to negate a statement?

Negating a statement means to disprove or invalidate it. It involves providing evidence or arguments that contradict or challenge the original statement.

3. Why is it important to negate statements?

Negating statements is important because it allows for critical thinking and encourages the examination of different perspectives. It also helps to prevent false or misleading information from being accepted as truth.

4. How can we determine if a statement needs to be negated?

A statement can be negated if it is not supported by evidence or logical reasoning. It can also be negated if it is biased, misleading, or contradicted by other credible sources.

5. What are some strategies for effectively negating a statement?

Some strategies for effectively negating a statement include conducting thorough research, using logical reasoning, providing evidence or counterarguments, and remaining open-minded to different perspectives. It is also important to communicate respectfully and avoid personal attacks.

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