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Infimum and supremum of empty set

  1. Jul 3, 2015 #1

    I can't wrap my mind around this:

    inf∅= ∞
    sup∅= - ∞

    Thank you in advance.
  2. jcsd
  3. Jul 3, 2015 #2


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    inf∅ is artificially defined to be ∞ so that inf will work well. Suppose we had defined inf∅ = 998 and we had a set with one element, 999. Then we want inf({999}) = 999. But since ∅ is also a subset of {999}, we would have inf({999}) = inf∅ = 998. The only way to avoid this problem is to make inf∅ greater than any possible number. So inf∅ = ∞. Similarly we have to define sup∅ smaller than any possible number. So sup∅ = -∞.

    In a sense, this is just getting ∅ out of the way of the calculation of inf and sup.
  4. Jul 3, 2015 #3
    Indeed, it gets ∅ out of the way!

    Thank you very much, FactChecker!
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