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Negating Complex Definitions

I understand how to negate simple statements, but how does one go about negating complex definitions such as:

Attachment 29765

I want to negate the quantifiers and then simply change the <'s to >'s, but it doesn't seem like this makes sense:

There exists an epsilon, for all delta, there exists an x, there exists a y...

How does one figure out what to negate/change?

Thanks for your help.

Re: Negating Complex Definitions

Hey divinelogos.

If you negate a "for-all" you get a "there exists a counter-example" and if you negate a "there exists" then you get "for all counter-examples". It's basically reversing the quantifiers but you also have to invert what the variable refers to.

Re: Negating Complex Definitions

Quote:

Originally Posted by

**divinelogos** I understand how to negate simple statements, but how does one go about negating complex definitions such as:

Attachment 29765
I want to negate the quantifiers and then simply change the <'s to >'s, but it doesn't seem like this makes sense:

There exists an epsilon, for all delta, there exists an x, there exists a y...

CORRECT
How does one figure out what to negate/change?

But note that $\displaystyle \neg \left( {z < b} \right) \equiv \left( {z \ge b} \right)$ and $\displaystyle \neg \left( {P \to Q} \right) \equiv \neg \left( {\neg P \vee Q} \right) \equiv \left( {P \wedge \neg Q} \right)$,

So we get $\displaystyle \exists \varepsilon \forall \delta \exists y\left[ {\left| {y - x} \right| \ge \delta \wedge \left| {f(y) - f(x)} \right| \ge \varepsilon } \right]$

Re: Negating Complex Definitions

Quote:

Hey divinelogos.

If you negate a "for-all" you get a "there exists a counter-example" and if you negate a "there exists" then you get "for all counter-examples". It's basically reversing the quantifiers but you also have to invert what the variable refers to.

Ok, I get changing "For all epsilon" to "There exists an epsilon", but what does "invert what the variable refers to" mean?

Re: Negating Complex Definitions

Quote:

Originally Posted by

**divinelogos** Ok, I get changing "For all epsilon" to "There exists an epsilon", but what does "invert what the variable refers to" mean?

$\displaystyle \forall x$ reads "**for all** $\displaystyle x$"

$\displaystyle \exists x$ reads "**there exists an** $\displaystyle x$"

$\displaystyle \neg \left( {\forall x[ P(x)]} \right) \equiv \left( {\exists x} \right)\left[ {\neg P(x)} \right]$

$\displaystyle \neg \left( {\exists x[ P(x)]} \right) \equiv \left( {\forall x} \right)\left[ {\neg P(x)} \right]$

Here is a clasic symbolic logic joke.

Read this $\displaystyle \forall\forall\exists\exists$. ANSWER "For every upside down A there is a backways E".