Logical Quantifiers: For all such that

[mliuzzolino]

Homework Statement

Negate the following statement:
For all x such that 0 < x < 1, f(x) <2 or f(x) > 5.

Homework Equations

I understand the universal quantifier is used as, "For all x, p(x)."
and the existential quantifier is used as, "There exists x such that p(x)."

I understand how to negate these alone; however, in this problem I am confused by "For all x such that..."

The Attempt at a Solution

Symbolic Statement:
\forall x \ni 0 < x < 1, f(x) < 2 or f(x) > 5.



Negation 1:

\exists x \ni 0 < x < 1, [f(x) = 2 or f(x) > 2] and [f(x) = 5 or f(x) < 5].


Negation 2:

\exists x \forall [x = 0 or x < 0] and [x = 1 or x > 1], [f(x) = 2 or f(x) > 2] and [f(x) = 5 or f(x) < 5].

 

[Joffan]

So, how would you negate "For all x, p(x)."?

 

[mliuzzolino]

Statement: For all x, p(x).

Negation: There exists an x such that ~p(x).

 

[pasmith]

Homework Statement

Negate the following statement:
For all x such that 0 < x < 1, f(x) <2 or f(x) > 5.

Homework Equations

I understand the universal quantifier is used as, "For all x, p(x)."
and the existential quantifier is used as, "There exists x such that p(x)."

I understand how to negate these alone; however, in this problem I am confused by "For all x such that..."

In strict technicality, you don't introduce an object without saying what it is. Thus you should never have "for all x" but "for all x in X". The negation of "for all x in X, P(x)" is "there exists x in X such that not P(x)".

In your example, assuming x is supposed to be real, $x \in X = \{y \in \mathbb{R}: 0 < y < 1\}$ which in interval notation is $x \in (0,1)$.

The Attempt at a Solution

Symbolic Statement:
\forall x \ni 0 < x < 1, f(x) < 2 or f(x) > 5.

Negation 1:

\exists x \ni 0 < x < 1, [f(x) = 2 or f(x) > 2] and [f(x) = 5 or f(x) < 5].

You need "x \in (0,1)" instead of "\ni 0 < x < 1", but otherwise this is correct. You can however simplify "[f(x) = 2 or f(x) > 2] and [f(x) = 5 or f(x) < 5]" to "$2 \leq f(x) \leq 5$".

 

[mliuzzolino]

Thanks pasmith!

That cleared up a lot for me. Much appreciated!

 

[HallsofIvy]

To answer your specific question, the negation of "for all x, P(x)" is "there exist x such that NOT P(x)"