Why do we define(by convention) that infimum of an empty set as \infty and supremum as -\infty?
May 3, 2009 #1 Edwinkumar 23 0 Why do we define(by convention) that infimum of an empty set as [tex]\infty[/tex] and supremum as [tex]-\infty[/tex]? Last edited: 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]?
May 3, 2009 #2 Hurkyl Staff Emeritus Science Advisor Gold Member 14,971 26 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.
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.
May 3, 2009 #3 matt grime Science Advisor Homework Helper 9,426 6 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.
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.