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    Can someone please help me with negating the following statement
    "There is  u \in \mathbb{R} such that for all  v \in \mathbb{R} there exists  w \in \mathbb{R} such that  u+v \geq w "
    What I have which is most probably wrong is
    "There is  u \not\in \mathbb{R} such that  \exists v \in \mathbb{R} for all  w \in \mathbb{R} such that  u+v < w ."
    Thanks.
    EDIT: Why on earth did I get neg rep?
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    (Original post by JBKProductions)
    Can someone please help me with negating the following statement
    "There is  u \in \mathbb{R} such that for all  v \in \mathbb{R} there exists  w \in \mathbb{R} such that  u+v \geq w "
    What I have which is most probably wrong is
    "There is  u \not\in \mathbb{R} such that  \exists v \in \mathbb{R} for all  w \in \mathbb{R} such that  u+v < w ."
    Thanks.
    Hmm. Went a bit awry on the first part of the sentence. "There is u in R..."
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    (Original post by SsEe)
    Hmm. Went a bit awry on the first part of the sentence. "There is u in R..."
    Thanks, so if I put  u \in \mathbb{R} in the first part of my sentence it would be correct or still wrong?
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    (Original post by JBKProductions)
    Thanks, so if I put  u \in \mathbb{R} in the first part of my sentence it would be correct or still wrong?
    No. You've managed to negate everything else except this. "There exists" becomes "for all".

    Also, make sure the thing actually reads like a sentence when you're done negating. ie, make sure the "such that"s are in the right places.
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    Let's say you have a statement of the following form: \exists u \in \mathbb{R} such that u satisfies property P. This means that there exists one real number satisfying P. The negation of this statement, in words, is: Every real number does not satisfy property P. Formally, this is: \forall u \in \mathbb{R}, u does not satisfy P

    So to negate a "there exists" statement, you change the exists to a for all and negate the property or condition. Similarly, the negation of a for all statement is an exists statement with the negated condition.

    Going along this line of thinking, I assume you just do the same thing for a statement with multiple exists or for all statements, but I'm not certain on that.
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    Thanks for the help.
    I've had another go at it and this is what I got
    "For all  u \in \mathbb{R} there exists  v \in \mathbb{R} such that for all  w \in \mathbb{R} , u+v < w "?
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    (Original post by JBKProductions)
    Thanks for the help.
    I've had another go at it and this is what I got
    "For all  u \in \mathbb{R} there exists  v \in \mathbb{R} such that for all  w \in \mathbb{R} , u+v < w "?
    Looks good to me.
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    (Original post by ghostwalker)
    Looks good to me.
    Thanks!
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    (Original post by JBKProductions)
    Thanks for the help.
    I've had another go at it and this is what I got
    "For all  u \in \mathbb{R} there exists  v \in \mathbb{R} such that for all  w \in \mathbb{R} , u+v < w "?
    That's what I'd come up with
 
 
 
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