Difference between infimum and minimum?

Question

Hi, what is the difference between infimum and minimum?

Answer 1

A "minimum" of a set is the smallest element of the set. An infinite set does not have to have a minimum, even if it is bounded.

An "infimum" is the greatest lower bound of a set--that is, the largest number k such that every element in the set is greater than or equal to k. A nonempty set of real numbers which has any lower bound at all must have an infimum.

For example:

The set {x | x >= 0} of nonnegative real numbers has 0 as its minimum (since 0 is in the set, and there is no element in the set less than 0). The number 0 is also the infimum (greatest lower bound) for this set; every number in the set is greater than or equal to 0, and no larger number has this property.

The set {x | x > 0} of positive real numbers also has 0 as its infimum. However, this set does not have a minimum. (0 is not contained in the set, so it can't be a minimum; and whatever number greater than 0 you might pick to be a minimum, there's always a number closer to 0 than that--there's no smallest positive real number.)

Some things to note:
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-If a set has a minimum, then it is always equal to the infimum of the set.
-Finite sets always have a minimum.
-"Maximum" and "supremum" are similar terms to measure the largest element of the set. They are analagous to "minimum" and "infimum."

Answer 2

A minimum has to be defined for the sequence, an infimum can be just a lower bound. For instance, the infimum of e^(-x) is 0, but 0 is not a minimum because e^(-x) is never 0.

Answer 3

First, infima and suprema pertain to SETS.

The infimum is the greatest lower bound for the set, and need not be in the set.

The minimum of a set is, of course, in the set.

Answer 4

The minimum is the smallest number in a set. This exists for all finite sets and all closed & bounded subsets of the Rreal Numbers -- but does not exist for all sets. For example, the set of all numbers greater than one and less than two has no minimum.

Infinum is the largest number that is less than or equal to all numbers of the set. In the above example, that is one. It need not be in the set.

Answer 5

Infimum

Answer 6

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