- #1

mliuzzolino

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## 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].