Infimun and supremum of empty set

  • Context: Graduate 
  • Thread starter Thread starter Edwinkumar
  • Start date Start date
  • Tags Tags
    Empty Set Supremum
Click For Summary
SUMMARY

The infimum of an empty set is defined as +∞ and the supremum as -∞ based on the definitions of least upper bound and greatest lower bound. Since there are no elements in the empty set, every real number qualifies as an upper bound, making the supremum -∞ by convention. Conversely, the infimum is +∞ because there are no lower bounds to consider. This understanding is crucial for mathematical rigor in set theory.

PREREQUISITES
  • Understanding of set theory concepts
  • Familiarity with the definitions of supremum and infimum
  • Basic knowledge of real numbers and their properties
  • Concept of upper and lower bounds
NEXT STEPS
  • Study the properties of upper and lower bounds in set theory
  • Explore the implications of empty sets in mathematical analysis
  • Learn about the completeness property of real numbers
  • Investigate the role of conventions in mathematical definitions
USEFUL FOR

Mathematicians, students of mathematics, and anyone interested in advanced set theory concepts will benefit from this discussion.

Edwinkumar
Messages
23
Reaction score
0
Why do we define(by convention) that infimum of an empty set as \infty and supremum as -\infty?
 
Last edited:
Physics news on Phys.org
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.
 
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.
 
Thanks Hurkyl and matt grime for your replies. Yes I got it now!
 

Similar threads

Undergrad Is there anything wrong with how the supremum of a set is written?
  • · Replies 5 ·
Replies
5
Views
2K
MHB Supremum and Infimum of Bounded Sets Multiplication
  • · Replies 1 ·
Replies
1
Views
2K
High School Question about set containing subsequential limits of bounded sequence
  • · Replies 6 ·
Replies
6
Views
2K
Undergrad Do I need induction to prove that this sequence is monotonic?
  • · Replies 1 ·
Replies
1
Views
2K
Can You Solve a Problem Using the Definition of Supremum?
  • · Replies 2 ·
Replies
2
Views
826
Undergrad Supremum & infimum of the set of all rational numbers
  • · Replies 2 ·
Replies
2
Views
5K
Undergrad How Do You Determine Supremum and Infimum Without Graphing?
  • · Replies 4 ·
Replies
4
Views
4K
Proving the Infimum and Supremum: A Short Guide for Scientists
  • · Replies 3 ·
Replies
3
Views
2K
MHB Infimum and Supremum of a Set (Need Help Finding Them)
  • · Replies 2 ·
Replies
2
Views
2K
MHB Finding the Infimum and Supremum
  • · Replies 4 ·
Replies
4
Views
1K