Logic, negation of a statement containing quantifiers

[SithsNGiggles]

Hi, I've got another answer I'd like checked. I'm pretty sure it works out, but I want to be certain.

Homework Statement

Write a sentence in everyday English that properly communicates the negation of each statement.

"Some differentiable functions are bounded."

Homework Equations

The Attempt at a Solution

First, I wrote the statement symbolically:
(\exists f(x) \in X) (f(x) \; \mbox{is bounded} \wedge f(x) \; \mbox{is differentiable}),
where I let X be the set of differentiable functions.

\neg (\exists f(x) \in X) (f(x) \; \mbox{is bounded} \wedge f(x) \; \mbox{is differentiable})

(\forall f(x) \in X) \neg (f(x) \; \mbox{is bounded} \wedge f(x) \; \mbox{is differentiable})

(\forall f(x) \in X) (f(x) \; \mbox{is not bounded} \vee f(x) \; \mbox{is not differentiable})

My question is, can I simplify this sentence to
(\forall f(x) \in X) (f(x) \; \mbox{is not bounded}),
since all f(x) \in X are differentiable and therefore cannot be differentiable?

I just want to make sure my reasoning works here. Thanks!

As for the English translation:
"Every differentiable function is not bounded."

 

[pasmith]

Hi, I've got another answer I'd like checked. I'm pretty sure it works out, but I want to be certain.

Homework Statement

Write a sentence in everyday English that properly communicates the negation of each statement.

"Some differentiable functions are bounded."

Homework Equations

The Attempt at a Solution

First, I wrote the statement symbolically:
(\exists f(x) \in X) (f(x) \; \mbox{is bounded} \wedge f(x) \; \mbox{is differentiable}),
where I let X be the set of differentiable functions.

So the condition "f(x) is differentiable" is equivalent to f(x) \in X and so redundant:
(\exists f(x) \in X) (f(x)\mbox{ is bounded})

\neg (\exists f(x) \in X) (f(x) \; \mbox{is bounded} \wedge f(x) \; \mbox{is differentiable})

(\forall f(x) \in X) \neg (f(x) \; \mbox{is bounded} \wedge f(x) \; \mbox{is differentiable})

(\forall f(x) \in X) (f(x) \; \mbox{is not bounded} \vee f(x) \; \mbox{is not differentiable})

My question is, can I simplify this sentence to
(\forall f(x) \in X) (f(x) \; \mbox{is not bounded}),
since all f(x) \in X are differentiable and therefore cannot be differentiable?

Yes: "P or false" is equivalent to P.

 

[HallsofIvy]

If you are defining X to be the set of all differentiable functions, then there is no need for "\and \text{f is differentiable}" in your original statement. With that definition of X, your statement is simply "(\exist f\in X)(f \text{is bounded})" and it negation is "(\all f\in X) (f \text{is not bounded})".

In any case, the negation of "some differentiable functions are bounded", in "every day English" is "no differentiable functions are bounded".