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Math Help - Express in words from symbols

  1. #1
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    Express in words from symbols

    Can anyone help with the following:

    Express in words the meaning of ∃xP(x) ∧ ∀x∀y ((P(x) ∧ P(y)) → (x = y))

    Here is what I came up with:

    For some P(x), and for every x and y, if P(x) and P(y), then x=y.

    But this doesn't make too much sense to me.
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  2. #2
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    ∃xP(x) just means there exists some x for which the property P holds.
    The following just means that there is only one element for which P holds.
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  3. #3
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    Quote Originally Posted by kturf View Post
    Can anyone help with the following:

    Express in words the meaning of ∃xP(x) ∧ ∀x∀y ((P(x) ∧ P(y)) → (x = y))

    Here is what I came up with:

    For some P(x), and for every x and y, if P(x) and P(y), then x=y.

    But this doesn't make too much sense to me.
    I find this statement strange. I haven't taken any formal logic classes but the operators are all common ones in math.

    I too think it means that for all x and y where P(x) and P(y) are true (maybe exist?) then x=y, implying that P(x)=P(y) implies x=y. I don't know if this is correct because many of the operators have multiple uses in different contexts. The main part that confuses me is the AND operator, ^. It could mean something else, lattice theory for example, which is why I find it strange. Also, y was never defined as an element but x was.

    I hope someone will explain this in detail for all of us...
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  4. #4
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    Quote Originally Posted by kturf View Post
    Can anyone help with the following:

    Express in words the meaning of ∃xP(x) ∧ ∀x∀y ((P(x) ∧ P(y)) → (x = y))

    Here is what I came up with:

    For some P(x), and for every x and y, if P(x) and P(y), then x=y.

    But this doesn't make too much sense to me.
    You can make any statement for the proposition P(x).

    For ∃xP(x), I can say there is at least a member, x, whose image is P(x).

    For ∀x∀y ((P(x) ∧ P(y)) → (x = y)), I can say for every two images who possess the same appearance, they are the same images of the same person, i.e, x and y are of the same person.

    This reminds me of the one-one function.

    Say: There is at least an x in the domain (meaning one or more). If for every x and y, f(x)=f(y), then x = y.

    P(x)^P(y) \rightarrow x=y could mean if x is P(x) and y is P(y), then x is y.
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