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Math Help - negation of statement.

  1. #1
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    negation of statement.

    Rewrite the following statement using iniversal and existential quantifiers.

    1.For all animals x, if x is a dog, then x has paws and x has a tail.

    the correct answer: There exists animal x, x is a dog and either x has no paws or x has no tail.




    ...However, this next statement is this true?

    p = Jim is tall and thin.

    ~p = Jim is not tall and thin.

    ..where only one part of the sentence is negated.

    if it is indeed true, then is this not correct: there exists animals x, if x is a dog, then x has paws and x does not have a tail.
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  2. #2
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    Quote Originally Posted by rcmango View Post
    ...However, this next statement is this true?

    p = Jim is tall and thin.

    ~p = Jim is not tall and thin.

    ..where only one part of the sentence is negated.
    p = Jim is tall and thin.

    ~p = Jim is either not tall or not thin (or both).

    The statement "Jim is not tall and thin" is ambiguous. I think most people would interpret it as "Jim is not (tall and thin)". But if you think that only one part of the sentence is being negated then presumably you are construing it as "Jim is (not tall) and thin".
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  3. #3
    Super Member angel.white's Avatar
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    Quote Originally Posted by rcmango View Post
    Rewrite the following statement using iniversal and existential quantifiers.

    1.For all animals x, if x is a dog, then x has paws and x has a tail.

    the correct answer: There exists animal x, x is a dog and either x has no paws or x has no tail.




    ...However, this next statement is this true?

    p = Jim is tall and thin.

    ~p = Jim is not tall and thin.

    ..where only one part of the sentence is negated.

    if it is indeed true, then is this not correct: there exists animals x, if x is a dog, then x has paws and x does not have a tail.
    try looking at the second statement like this:
    For all people, x, if x is Jim, then x is tall and thin.
    Let J_{(x)} be "x is Jim"
    Let T_{(x)} be "x is Tall and thin"

    \forall_x [J_{(x)}\rightarrow T_{(x)}]

    So the negation is:
    \neg\forall_x [J_{(x)}\rightarrow (T_{(x)})]
    Which can be read as "It is not the case that every Jim is tall and thin."
    And since there is only one Jim (of which we speak) we could simplify it to
    "It is not the case that Jim is tall and thin"
    Which simplifies to
    "Jim is not tall and thin"

    Now, I think you are getting confused because this statement seems to you to say that "Jim is not tall and Jim is not thin" but that is not the case. Due to DeMorgan's law, \neg(tall \wedge thin) \equiv \neg tall \vee \neg thin
    If you have difficulty seeing this, try drawing a truth table with the variables A and H, and the headings A\wedge H, \neg(A \wedge H), \neg A \vee \neg H and \neg(A\wedge H) \equiv \neg A \vee \neg H. If you do it right, you should see they are logically equivalent.

    So you could also read it as "Jim is not thin or Jim is not tall"
    So really, they are saying the same thing in both sentences, it just didn't appear that way at first.
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