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Infimum of subsets in R

  1. Jun 25, 2014 #1
    1. The problem statement, all variables and given/known data

    If a is both the infimum of A[itex]\subseteq \mathbb{R}[/itex] and of B[itex]\subseteq \mathbb{R}[/itex] then a is also the infimum of A[itex]\cap[/itex]B

    Is this statement true or false? If true, prove it. If false, give a counterexample.

    2. Relevant equations



    3. The attempt at a solution

    I think it's true because let's say A={1,2,3,4} and B={1,2,3} then A[itex]\cap[/itex]B = {1,2,3}.

    Then inf {A}= 1 and inf {B} = 1.
    And inf {A[itex]\cap[/itex]B} = 1.

    However, I think it's false because, and correct me if I'm wrong, the infimum doesn't necessarily have to belong to the subsets A nor B to be an infimum. The infimum can also be a value outside of those sets. Which would imply that the infimum of A and B doesn't have to be equal to the infimum of A[itex]\cap[/itex]B.
     
  2. jcsd
  3. Jun 25, 2014 #2

    pasmith

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    What happens if [itex]A \cap B[/itex] is empty? Nothing in the problem statement says that they have to intersect, so long as they have the same infimum which, as you point out, does not have to be a member of either A or B.

    Is it possible to have two subsets [itex]A[/itex] and [itex]B[/itex] with [itex]\inf A = \inf B[/itex] and [itex]A \cap B = \varnothing[/itex]?
     
  4. Jun 25, 2014 #3
    Hm, I don't think that last part is possible. Both sets have something in common, i.e. the infimum, which would imply [itex]A \cap B[/itex] is not empty. Is my reasoning correct?
     
  5. Jun 25, 2014 #4

    verty

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    Have you heard of Zeno's paradox (the well-known one I mean)?
     
  6. Jun 25, 2014 #5
    Yes. Why?
     
  7. Jun 25, 2014 #6

    HallsofIvy

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    No it is not. Let A be the set of all positive rational numbers. It's infimum is 0. Let B be the set of all positive irrational numbers. Its infimum is also 0. But their intersection is empty.
     
  8. Jun 25, 2014 #7
    So this would fit as a counterexample because you've found the exact same infimum for set A and set B, i.e. 0, and this infimum does not equate to the infimum of their empty intersection?
     
  9. Jun 25, 2014 #8

    Zondrina

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    Indeed, ##inf(ø) = ∞## and ##sup(ø) = -∞##.
     
  10. Jun 25, 2014 #9
    Thanks! Unrelated question : Any advice to someone learning this on his own? I love physics and I know I have to grind through the mathematical details because they matter too but sometimes I get a bit frustrated if I don't immediately get the answer correct.
     
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