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State the trichotomy law formally

  1. Jan 22, 2017 #1
    1. For arbitrary real numbers a & b, exactly one of the three relations hold:
    a < b, a > b, a = b.


    How do I state this more formally while also being correct?


    2. The attempt at a solution
    a, b ∈ ℝ ( (a < b) ⊕ ( a > b ) ⊕ ( a = b) )

    From this I made a truth table 2^3 entries long, and what we need is for the solution to only be true when exactly one relation is true and the rest false. The xoring works logically for 7 of the 8 entries, but fails when all three values are true. One solution off the top of my head is to simply add the following snippet ( ∧ ¬( (a < b) ∧ ( a > b ) ∧ ( a = b) )). That seems really ugly huh? But is that what the statement (exactly one of the three relations hold) turns into?

    I am a self learner, anyone want to help me improve? Thank you!
     
  2. jcsd
  3. Jan 22, 2017 #2
    I think, You can also use this equivalent one:
    [tex] (a \wedge \bar{b} \wedge \bar{c} ) \vee (\bar{a} \wedge b \wedge \bar{c}) \vee (\bar{a} \wedge \bar{b} \wedge c ) [/tex]

    where a,b,c are the three statements.
     
  4. Jan 22, 2017 #3
    Nice BD! I really like this, I feel there's something really beautiful about your formula. Something about the balancing of truthness between all three statements. One being true, the other two are 'weighed' with it, and the other two ones must be the inverse of the true one for the 'weighing' to come out positive. All three are tested together to find the one of three statements that are true, and because of the nature of the test, if any two are true, there's a contradiction. Did you just think of this on the spot?
     
  5. Jan 22, 2017 #4
    No, I played around with it on paper, then double checked the logical equivalence on wikipedia so I don't look like an annoying fool!!! I have bought schaums outlines of logic and its sitting around on my desk :sorry:, I am just trying to apply what I know so I don't forget in the long run! =) I think logical equivalences are quite interesting and some quite beautiful and unexpected.
     
  6. Jan 23, 2017 #5

    Mark44

    Staff: Mentor

    As you have it above, it looks perfectly fine. Is there some reason you wanted to represent the trichotomy using symbolic logic?
     
  7. Jan 23, 2017 #6
    1) Just a pure joy from symbolic manipulation ( my brain is weird :D )
    2) For more complicated statements, human language is ambiguous, so being able to translate a theorem from English to Logic is invaluable for my own type of thinking and understanding.
    2a) I don't have to write as much
    2b) It's easier to read the argument when it's a concise line of symbols
    3) Some books state their model using a combination of set theory and logic instead of english and I want to be prepared
    4) It's fun learn something new.
    5) Maybe someday I'll have the luxury of exploring math foundations
     
  8. Jan 23, 2017 #7

    Mark44

    Staff: Mentor

    The three inequalities you started with are symbols, and are completely unambiguous.
    Really? This -- ##(a \wedge \bar{b} \wedge \bar{c} ) \vee (\bar{a} \wedge b \wedge \bar{c}) \vee (\bar{a} \wedge \bar{b} \wedge c )## is easier to write than this -- x < y, or x = y, or x > y?

    Not to mention that what Bipolar Demon actually wrote was this (unrendered LaTeX):
    (a \wedge \bar{b} \wedge \bar{c} ) \vee (\bar{a} \wedge b \wedge \bar{c}) \vee (\bar{a} \wedge \bar{b} \wedge c )
    x < y, or x = y, or x > y is pretty easy to read.
    The problem is that a few people overuse symbolism to the point that it obfuscates the point they are trying to make. If there's a reason for using symbols instead of a prose explanation, fine, but using symbolism for its own sake should be avoided, IMO.
    No argument there.
     
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