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Math Help - Logical statement translation?

  1. #1
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    Question Logical statement translation?

    I know that for this statement :

    "this equation x=1 has 2 different solutions only"

    the translation will be :

    ƎxƎy( x=1 ʌ y=1 ʌ x≠y ʌ ∀z( z=1 → (z=xVy=y)).

    now it might sound a bit dumb to ask, but what if the equation would be x-1=2x ?

    thanks.

    p.s - feel free to correct my freely translated terms, ex : I'm not sure if you say translate or formalize... it would help me be clearer for further questions )
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  2. #2
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    Write as x^2-2x=1 and do exactly as you did above.
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  3. #3
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    even though the right side has 2x (which is part of the equation?) ?
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  4. #4
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    \left( {\exists x} \right)\left( {\exists y} \right)\left( {\forall z} \right)\left[ {x \ne y \wedge \left( {x^2  - 1 = 2x} \right) \wedge \left( {y^2  - 1 = 2y} \right) \wedge \left( {\left( {z^2  - 1 = 2z} \right) \to z = x \vee z = y} \right)} \right]
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  5. #5
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    ohh oops lol, I just now got your last post :P, thanks a lot.
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  6. #6
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    Plato is probably right. In addition, if I wanted to make a statement about the number of roots, I would not use the phrase "the equation x=1 has 2 different solutions only". Sometimes people say, "There is one and only one solution", where by "there is one" they mean "at least one", and this is not sufficiently strong, so they add the "only one" part. I am wondering if by "only one" they mean "at most one" and if this part can be used alone, i.e., if one can say, "There is only one solution" meaning that there is at most one solution.

    So, to make the statement unambiguous, I would say, "The equation has exactly two solutions".
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  7. #7
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    Ah, the vagueness of language.
    One of my favorite mathematical stories involves Solomon Lefschetz’s and E.H. Moore. Moore was giving a guest lecture at Princeton. He began his talk “Let a be a point and b be a point”. Leftschetz shouted out “But why don’t you just say ‘let a and b be points’?” Moore replyed “Because a could be b”. Leftschetz got up and left the room.
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  8. #8
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    so my question should've been phrased like : The equation has exactly two different solutions?

    btw, how would the translation look if it said : The equation has at most two different solutions?

    thanks.
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  9. #9
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    Quote Originally Posted by Plato View Post
    Ah, the vagueness of language.
    One of my favorite mathematical stories involves Solomon Lefschetz’s and E.H. Moore. Moore was giving a guest lecture at Princeton. He began his talk “Let a be a point and b be a point”. Leftschetz shouted out “But why don’t you just say ‘let a and b be points’?” Moore replyed “Because a could be b”. Leftschetz got up and left the room.
    that's a really nice story that justify even more why I don't like any written math questions.
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  10. #10
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    btw, how would the translation look if it said : The equation has at most two different solutions?
    You can say, for every three solutions x y z, x = y or x = z or y = z.
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  11. #11
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    Quote Originally Posted by emakarov View Post
    You can say, for every three solutions x y z, x = y or x = z or y = z.
    Thank you.

    do you know of any good book/site that explains this specific topic?

    I tried googling "logical translations" and could find anything relative, maybe I'm not looking for the right thing?

    thanks again.
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  12. #12
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    The best book I've found is 'Logic: Techniques Of Formal Reasoning' by Kalish, Montague, and Mar.

    Indeed it has an especially good treatment of the subject of specific finite quantities expressed in first order logic with identity.

    Anyway, it's pretty simple. For any natural number n, and any exprssible predicate P, there is an obvious first order formula to express "there are exactly n number of x such that P holds for x." And it's not much harder to expand this for k-place predicates for any k.
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  13. #13
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    Quote Originally Posted by emakarov View Post
    You can say, for every three solutions x y z, x = y or x = z or y = z.
    wait, what about the different part?

    maybe it could be : x≠y ʌ (z → x=z V y=z) ?

    EDIT : ^^ ohh man, now it's just the same as the first one I think :/

    EDIT : or maybe something like this ƎxƎy∀z( x=1 V y=1 ʌ x≠y ʌ ( z=1 → (z=xVy=y)).
    Last edited by Laban; October 30th 2010 at 12:37 PM.
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  14. #14
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    The propositions "for all x y z, P(x) /\ P(y) /\ P(z) -> x = y \/ x = z \/ y = z" and "There exist at most two different x's such that P(x)" are logically equivalent. Indeed, in interpretations where P is true on zero, one or two distinct domain elements, both propositions are true, and on other interpretations, both are false.
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