# Negating Complex Definitions

• Nov 18th 2013, 07:54 PM
divinelogos
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?

• Nov 18th 2013, 11:32 PM
chiro
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.
• Nov 19th 2013, 03:14 AM
Plato
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 $\neg \left( {z < b} \right) \equiv \left( {z \ge b} \right)$ and $\neg \left( {P \to Q} \right) \equiv \neg \left( {\neg P \vee Q} \right) \equiv \left( {P \wedge \neg Q} \right)$,

So we get $\exists \varepsilon \forall \delta \exists y\left[ {\left| {y - x} \right| \ge \delta \wedge \left| {f(y) - f(x)} \right| \ge \varepsilon } \right]$
• Nov 20th 2013, 01:01 PM
divinelogos
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?
• Nov 20th 2013, 01:17 PM
Plato
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?

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

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

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

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

Here is a clasic symbolic logic joke.

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