Negating a universal quantifier: ∀? ∈ ]0,1[ ∶ ?2 < ?

[MathLearner2016]

Negating a universal quantifier: ∀? ∈ ]0,1[ ∶ ?2 < ?

I've never seen a problem like this before:

∀? ∈ ]0,1[ ∶ ?2 < ?

I think it reads: For all members x in the set "]0,1[" applies "x2 < x". How do I negate this without using ¬ (negation symbol)?

 

[pka]

I've never seen a problem like this before:
∀? ∈ ]0,1[ ∶ ?2 < ? How do I negate this without using ¬ (negation symbol)?

First note that is rarely used notation for an open interval: \(\displaystyle (0,1)=~]0,1[~=\{x:0<x<1\}\)
It is true that \(\displaystyle \forall x\in (0,1)[x^2<x]\) its negation is \(\displaystyle \exists y\in (0,1)[y^2\ge y]\)

 

[MathLearner2016]

First note that is rarely used notation for an open interval: \(\displaystyle (0,1)=~]0,1[~=\{x:0<x<1\}\)
It is true that \(\displaystyle \forall x\in (0,1)[x^2<x]\) its negation is \(\displaystyle \exists y\in (0,1)[y^2\ge y]\)

Where did the y come from?

 

[stapel]

Where did the y come from?

It's just another generic element in the set. You could use "x" instead, if you like.

 

[pka]

It's just another generic element in the set. You could use "x" instead, if you like.

Actually though it is rather technical, it is generally agree the different variables are used because "x" is free in the universal and "y" is bound in the existential.