Can an empty set have an infimum?

Formally there is no infimum or supremum of the empty set since both of these, when they exist, are defined to be Real Numbers.

What is the Supremum and infimum of empty set respectively?

If we consider subsets of the real numbers, then it is customary to define the infimum of the empty set as being ∞. This makes sense since the infimum is the greatest lower bound and every real number is a lower bound. So ∞ could be thought of as the greatest such. The supremum of the empty set is −∞.

How do you find the infimum and supremum of a set?

Suppose that A ⊂ R is a set of real numbers. If M ∈ R is an upper bound of A such that M ≤ M′ for every upper bound M′ of A, then M is called the supremum of A, denoted M = sup A. If m ∈ R is a lower bound of A such that m ≥ m′ for every lower bound m′ of A, then m is called the or infimum of A, denoted m = inf A.

Does an empty set have a maximum?

As an example, the set [0,1) with the standard order on R has a supremum (1) but no maximum. As for this particular question: it’s rather trivial, since a maximum of A is necessarily an element of A, but nothing is an element of the empty set.

Which is the supremum of the empty set?

The supremum of the empty set is − ∞. Again this makes sense since the supremum is the least upper bound. Any real number is an upper bound, so − ∞ would be the least.

Which is the infimum of the empty set?

If we consider subsets of the real numbers, then it is customary to define the infimum of the empty set as being \\infty. This makes sense since the infimum is the greatest lower bound and every real number is a lower bound. So \\infty could be thought of as the greatest such.

How to show if the supremum and the infimum exist?

Q: Show that if the supremum and the infimum exist, they must be unique. Let S be a set and assume that b is a supremum for S. Assume as well that c is also supremum for S and we need to show b = c. Since c is supremum, it is an upper bound for S. Since b is supremum, then it is the least upper and thus b ≤ c.

Which is the supremum of a bounded set?

A set is bounded if it is bounded both from above and below. The supremum of a set is its least upper bound and the infimum is its greatest upper bound. Definition 2.2. Suppose that A ⊂ R is a set of real numbers. If M ∈ R is an upper bound of A such that M ≤ M′for every upper bound M′of A, then M is called the supremum of A, denoted M = supA.