Difference between supremum and maximum

Referring to this lecture , I want to know what is the difference between supremum and maximum. It looks same as far as the lecture is concerned when it explains pointwise supremum and pointwise maximum

Answer

A maximum of a set must be an element of the set. A supremum need not be.

Explicitly, if X is a (partially) ordered set, and S is a subset, then an element s0 is the supremum of S if and only if:

  1. ss0 for all sS; and
  2. If tX is such that st for all sS, then s0t.

By contrast, an element m is the maximum of S if and only if:

  1. sm for all sS; and
  2. mS.

Note that if S has a maximum, then the maximum must be the supremum: indeed, if tX is such that st for all sS, then in particular mS, so mt, proving that m satisfies the conditions to be the supremum.

But it is possible for a set to have a supremum but not a maximum. For instance, in the real numbers, the set of all negative numbers does not have a maximum: there is no negative number m with the property that nm for all negative numbers n. However, the set of all negative numbers does have a supremum: 0 is the supremum of the set of negative numbers. Indeed, a0 for all negative numbers a; and if ab for all negative numbers a, then 0b.

The full relationship between supremum and maximum is:

  1. If S has a maximum m, then S also has a supremum and in fact m is also a supremum of S.
  2. Conversely, if S has a supremum s, then S has a maximum if and only if sS, in which case the maximum is also s.

In particular, if a set has both a supremum and a maximum, then they are the same element. The set may also have neither a supremum nor a maximum (e.g., the rationals as a subset of the reals). But if it has only one them, then it has a supremum which is not a maximum and is not in the set.

Attribution
Source : Link , Question Author : user31820 , Answer Author : Arturo Magidin

Leave a Comment