1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Definition of Strong and Weak form of a theorem

  1. Mar 14, 2009 #1

    JFo

    User Avatar

    Could someone explain to me what it means for a theorem to be a strong(er) form or a weak(er) form of another theorem?

    I've heard these terms used over and over, but never bothered to ask. If I had to guess at a definition, I'd say that if q is a theorem then we say p is stronger if p implies q. Similarly p is weaker if q implies p. Am I close?
     
  2. jcsd
  3. Mar 14, 2009 #2

    matt grime

    User Avatar
    Science Advisor
    Homework Helper

    Yes. Generally - aren't there always exceptions in mathematics.

    As a general rule the strong version will need more hypotheses than the weak version, but correspondingly prove a result that is stronger (and implies the weaker version).

    This is not always the case, as there is at least on situation where the strong and weak version actually are equivalent: induction and strong induction are the same, but phrased differently so that one seems like a stronger (more powerful) result.
     
  4. Mar 14, 2009 #3

    JFo

    User Avatar

    Thanks. I know it's mostly semantics, but I couldn't find a definition, or even a description, anywhere.
     
  5. Mar 14, 2009 #4

    JFo

    User Avatar

    So if we have a theorem, and are then able to show that we can drop one of the hypotheses (or replace it with a weaker hypothesis) and still obtain the same result, would that be considered "strengthening" or "weakening"

    Intuitively I would think strengthening, but then I don't know how that fits in with the quasi-definition given above since we have less hypotheses but the result remains unchanged. Or is this a different meaning of strong/weak entirely?
     
  6. Mar 14, 2009 #5
    The theorem that assumes less in the hypothesis is "stronger" or "more general". I wouldn't worry too much about what these terms exactly mean though, since they don't carry much information.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook