Infimun and supremum of empty set

Edwinkumar · May 3, 2009

Why do we define(by convention) that infimum of an empty set as [tex]\infty[/tex] and supremum as [tex]-\infty[/tex]?

Hurkyl · May 3, 2009

It's not a convention -- it follows directly from the definition of the supremum as the least upper bound and the infimum as the greatest lower bound.

matt grime · May 3, 2009

Remember that we say M is an upper bound for X if for all x in X... so if X is the empty set then this is never true. Now, "false implies true is true", i.e. all possible real numbers are upper bounds for the the empty set.

Edwinkumar · May 5, 2009

Thanks Hurkyl and matt grime for your replies. Yes I got it now!

Infimun and supremum of empty set

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Infimun and supremum of empty set

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