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Math Help - logical statements, does this imply this?

  1. #1
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    logical statements, does this imply this?

    let P(p,q) be a statement about a pair of natural numbers p and q.

    does

    "there exists a p such that for all q, P(p,q)"

    imply

    "for all q, there exists a p such that P(p,q)"

    i have tried a formal argument but i haven't had much experience with this stuff.
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  2. #2
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    Re: logical statements, does this imply this?

    i apologise if this is wrong, as i haven't done discrete math...but doesn't it follow by definition?

    eg:

    "there exists a p such that for all q, P(p,q). call this P*.

    by definition from the previous statement, for all q we have P(p*,q) which is the required result.
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  3. #3
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    Re: logical statements, does this imply this?

    that's exactly what i thought, what confused me was that on the worksheet they wanted an example of a statement for which the first was true but not the second. :S
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  4. #4
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    Re: logical statements, does this imply this?

    Quote Originally Posted by SpringFan25 View Post
    "there exists a p such that for all q, P(p,q). call this P*.

    by definition from the previous statement, for all q we have P(p*,q) which is the required result.
    You are right, though it should be p*, not P*. When one has both p and P, one has to be careful about uppercase and lowercase letters.

    So, if one p satisfies P for all q, then for every q we can pick that particular p. Note that the converse is false: if for every q we have its own p that makes P true, in general we can't select a single p that would make P true for all q.
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  5. #5
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    Re: logical statements, does this imply this?

    logical statements, does this imply this?-numebrs.png

    here is the problem (part (b) )
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  6. #6
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    Re: logical statements, does this imply this?

    There must be a mistake in part (b) because there is no such property P, as we discussed.
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