Infimum and supremum of empty set

[strobeda]

Hello,

I can't wrap my mind around this:

inf∅= ∞
sup∅= - ∞

Thank you in advance.

 

[FactChecker]

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.

 

[strobeda]

Indeed, it gets ∅ out of the way!

Thank you very much, FactChecker!