SOLUTION: a number set has no supremum or it has a supremum which is infinity . does the two statement same ??? if a set is S={1} what will be it's supremum and infimum also show it's all

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Question 975490: a number set has no supremum or it has a supremum which is infinity . does the two statement same ??? if a set is S={1} what will be it's supremum and infimum also show it's all upper bounds and lower bounds too.
Answer by solver91311(24713) About Me  (Show Source):
You can put this solution on YOUR website!


A set cannot have a supremum that is infinity. A supremum is a number, whereas infinity is not. If S = {1}, and S is a subset of T, then T must contain the element 1, and then 1 is the smallest element of T that is greater than or equal to all elements of S. Supremum and Least Upper Bound being equivalent terms, 1 is also the Least Upper Bound. Similarly, 1 is the infimum or greatest lower bound.

Note that if set S was defined as all real numbers in the open interval and T is the set of all real numbers, neither the supremum of S nor the infimum of S are actually elements of S. Here the supremum is 1 and the infimum is 0.

John

My calculator said it, I believe it, that settles it