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Math Help - Universal Quantifier Question

  1. #1
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    Universal Quantifier Question

    Is one of these incorrect?
    Also, I often see in proofs that "x in Z" and later they say "but x is arbitrary" is it wrong to use an existential quantifier at the beginning, or is it implied?


    1.
    (Universal Quantifier) m in Z is even <=> (existential quantifier) k in Z such that 2k = m


    2. m in Z is even <=> (existential quantifier) k in Z such that 2k = m
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  2. #2
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    \displaystyle \forall means "for all".

    So \displaystyle \forall x \in \mathbf{Z} means "for all x that are integers..."

    What's wrong with that?
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  3. #3
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    So there's no difference between saying:

    (universal quantifier) x in Z

    x in Z
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  4. #4
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    2. m in Z is even <=> (existential quantifier) k in Z such that 2k = m
    This statement has a free variable m, i.e., a variable that is not bound by any quantifier. Generally, formulas with free variables are not propositions, i.e., not something that is either true or false because for each value of a free variable such formula may have its own truth value. For this particular formula, it happens that it is true for all values of m.

    Quote Originally Posted by Noxide View Post
    Also, I often see in proofs that "x in Z" and later they say "but x is arbitrary" is it wrong to use an existential quantifier at the beginning, or is it implied?
    A statement "P(x) holds for an arbitrary x in Z", where P(x) is some expression, is equivalent to "For all x in Z, P(x)".
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