Results 1 to 2 of 2

Math Help - Question about nested quantifiers.

  1. #1
    Newbie
    Joined
    Dec 2012
    From
    United States
    Posts
    3

    Question about nested quantifiers.

    Can someone explain the difference between this:

    ∀x((x =0)→∃y(xy=1))

    .. and this:

    ∀x∃y((x =0)→(xy=1))

    The top one is in my book. The domain is all real numbers. I understand that it's saying that every real number other than zero has a multiplicative inverse, but I don't understand why it can't be written as the second one? Thank you.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Oct 2009
    Posts
    5,537
    Thanks
    778

    Re: Question about nested quantifiers.

    You probably mean x\ne0 and not x = 0 since you write, "every real number other than zero."

    The two formulas are equivalent. The second one has an advantage that it is in the prenex normal form. However, the first formula better reflects the meaning of the statement. It guarantees the existence of some y such that xy = 1 only after one provides a proof of x\ne0. The second formula requires producing some y just on the basis of a given x. In particular, even for x = 0 the interpretation has to provide some y. Later it turns out that this y does not matter because the assumption of the quantifier-free part (matrix) is false, so the matrix is true.

    If you consider the two formulas as types of computer programs, the first program takes an x and an evidence that x\ne0 and returns a y and an evidence that xy = 1. The second program takes an x and immediately returns a y such that if later an evidence of x\ne0 is given, then an evidence of xy = 1 is returned. The first program is arguably more reasonable because it returns something only when it has all necessary inputs provided.

    However, once again, these are subtleties. In the usual classical logic, both formulas are equivalent.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 1
    Last Post: August 26th 2009, 08:04 AM
  2. Nested Quantifiers
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: February 23rd 2009, 03:41 PM
  3. Nested Quantifiers
    Posted in the Discrete Math Forum
    Replies: 5
    Last Post: October 6th 2008, 11:53 PM
  4. Quantifiers question
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: January 31st 2008, 10:25 AM
  5. nested quantifiers
    Posted in the Discrete Math Forum
    Replies: 0
    Last Post: June 19th 2007, 08:48 AM

Search Tags


/mathhelpforum @mathhelpforum