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Math Help - Inverse (statements)

  1. #1
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    Inverse (statements)

    Is the inverse of the following compound statement correct?


    Statement:

    (For all fruit x, x is red) => [ (all houses are blue) or (some chickens are white) ]


    Inverse:
    (For some fruit x, x is not red) => [ (some houses are not blue) and (no chickens are not white) ]


    My confusion comes from the hypothesis of the implication. It is of the form
    [ (universal quantifier)(subject), (subject)(verb)(predicate) ]. I know a statement must be of the form [ (subject)(verb)(predicate) ] and also that the negation of [ (universal quantifier)(statement) ] is [ (existential quantifier)(negation of statement) ] but what is the negation of [
    (universal quantifier)(subject), (subject)(verb)(predicate) ]?
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  2. #2
    Member oldguynewstudent's Avatar
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    Quote Originally Posted by Noxide View Post
    Is the inverse of the following compound statement correct?


    Statement:

    (For all fruit x, x is red) => [ (all houses are blue) or (some chickens are white) ]


    Inverse:
    (For some fruit x, x is not red) => [ (some houses are not blue) and (no chickens are not white) ]


    My confusion comes from the hypothesis of the implication. It is of the form
    [ (universal quantifier)(subject), (subject)(verb)(predicate) ]. I know a statement must be of the form [ (subject)(verb)(predicate) ] and also that the negation of [ (universal quantifier)(statement) ] is [ (existential quantifier)(negation of statement) ] but what is the negation of [
    (universal quantifier)(subject), (subject)(verb)(predicate) ]?
    \neg{\exists}P(x)= \forall{x} \neg{P(x)}

    So it should be:

    (For some fruit x, x is not red) => [ (some houses are not blue) and (all chickens are not white) ]
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  3. #3
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    You are incorrect on the 'chicken one'.
    The negation of "some chickens are white" is "no chicken is white".
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  4. #4
    Member oldguynewstudent's Avatar
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    Quote Originally Posted by Plato View Post
    You are incorrect on the 'chicken one'.
    The negation of "some chickens are white" is "no chicken is white".
    Thank you, Plato. However, I don't see the difference between "no chicken is white" and "all chickens are not white".

    Could you elaborate on this? Thanks.
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  5. #5
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    Actually they are saying the same thing.
    However in this case it a matter of history and style.
    Based on what is known as the square of opposition.
    It has deep roots in the history of logic.

    There are four types of statements.
    Universal, positive statements are labeled A: All P is Q.
    Universal, negative statements are labeled E: No P is Q.
    Existential, positive statements are labeled I: Some P is Q.
    Existential, negative statements are labeled O: Some P is not Q.
    Statements A & O are negations of each other; as are E & I.

    With that background in mind, we generally want students to use the classical form.
    Last edited by Plato; May 21st 2010 at 02:38 PM.
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