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Lower Bound of given pair.

  1. Mar 10, 2015 #1
    Since I'm not sure if posting assignment questions is allowed, I'm just going to ask specific questions just to be safe.

    1. The problem statement, all variables and given/known data


    Find all the lower bounds of given pair. Say (a, b) in T.

    2. Relevant equations

    Proof for greatest lower bound:

    ∀g,a,b ∈ T ⇔ ( g ≺ a) ^ (g ≺ b) ^ ( ∀l ∈ T [ (l ≺ a ) ^ ( l ≺ b)] ⇒ (l ≺ g)

    by "≺", I meant Partial Order, ≤

    3. The attempt at a solution

    Since g are ordered before a and b. Can we assume for pair {2, 4}, g will be equivalent to

    {empty set}, {1,0}, {1,1}, {1,2}, {1,3}, {1,4}, {2, 1}, {2,2}, {2,3}, {2,4},

    I am not sure if I'm on the right track, but that's what it seems to be according to the proof.
     
  2. jcsd
  3. Mar 11, 2015 #2
    In part 1 you write (a, b) which could be an ordered pair or an open interval in ℝ but then you use set notation in part 3. Those could have greatest lower bounds if there's a partial ordering defined on them but I think I need more info. Part 2 makes sense though, a lower bound different from g is less than g but what are they & how are they being compared?

    edit: for example with {2, 4} ⊆ (ℕ, ≤) (meaning the set is ℕ & its partial order is ordinary < or =) the greatest lower bound is 2 & the other lower bound is 1. If {2, 4} ⊆ (ℤ, ≤) then the glb is still 2 but the other lower bounds are all the integers less than 2 .
     
    Last edited: Mar 11, 2015
  4. Mar 11, 2015 #3
    So if the proof for lower bound was applied to the pair {2,4} in T.

    ∀g,2,4 ∈ T ⇔ ( g ≺ 2) ^ (g ≺ 4) ^ ( ∀l ∈ T [ (l ≺ 2 ) ^ ( l ≺ 4)] ⇒ (l ≺ g)

    In other words, g comes before 2 and 4 while l comes before g. I'm trying to compute the lower bound of the given pair. I also forgot to mention that T = ( X8, ≺ ) where ≺ is defined on X8 = { 1, 2, 3, 4,5, 6, 7, 8} by the rule x ≺ y ⇔ 5x ≤ 3y.

    Therefore I was able to produce the pairs:

    {empty set}, {0,0}, {0,1}, {0,2}, {0,3,}, {0,4}, {1,2}, {1,3}, {1,4}, {2,4},

    Each of pairs produced go before their respective elements.

    Is this correct?
     
  5. Mar 12, 2015 #4
    Now that I understand how you got those pairs, once you know that ##2 \prec 4## because 5×2 ≤ 3×4 you don't need to compare 4 with any lesser elements because it obviously isn't a lower bound for {2, 4}. The way I understood the problem was that you want ##x \in X_{8}## such that ##x \prec 2##, in other words x such that 5×x ≤ 3×2 = 6 which I think you got but you need to give the elements (not subsets) of ##X_{8}## that are lower bounds of {2, 4}, which are 0, 1 & 2. The glb is 2 because it's the lower bound which is greater than the other ones.
     
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