- #1
SithsNGiggles
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Hi, I've got another answer I'd like checked. I'm pretty sure it works out, but I want to be certain.
Write a sentence in everyday English that properly communicates the negation of each statement.
"Some differentiable functions are bounded."
First, I wrote the statement symbolically:
[itex](\exists f(x) \in X) (f(x) \; \mbox{is bounded} \wedge f(x) \; \mbox{is differentiable})[/itex],
where I let [itex]X[/itex] be the set of differentiable functions.
[itex]\neg (\exists f(x) \in X) (f(x) \; \mbox{is bounded} \wedge f(x) \; \mbox{is differentiable})[/itex]
[itex](\forall f(x) \in X) \neg (f(x) \; \mbox{is bounded} \wedge f(x) \; \mbox{is differentiable})[/itex]
[itex](\forall f(x) \in X) (f(x) \; \mbox{is not bounded} \vee f(x) \; \mbox{is not differentiable})[/itex]
My question is, can I simplify this sentence to
[itex](\forall f(x) \in X) (f(x) \; \mbox{is not bounded})[/itex],
since all [itex]f(x) \in X[/itex] 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."
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:
[itex](\exists f(x) \in X) (f(x) \; \mbox{is bounded} \wedge f(x) \; \mbox{is differentiable})[/itex],
where I let [itex]X[/itex] be the set of differentiable functions.
[itex]\neg (\exists f(x) \in X) (f(x) \; \mbox{is bounded} \wedge f(x) \; \mbox{is differentiable})[/itex]
[itex](\forall f(x) \in X) \neg (f(x) \; \mbox{is bounded} \wedge f(x) \; \mbox{is differentiable})[/itex]
[itex](\forall f(x) \in X) (f(x) \; \mbox{is not bounded} \vee f(x) \; \mbox{is not differentiable})[/itex]
My question is, can I simplify this sentence to
[itex](\forall f(x) \in X) (f(x) \; \mbox{is not bounded})[/itex],
since all [itex]f(x) \in X[/itex] 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."
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