Supremum = least upper bound, anything > supremum?

In summary, the supremum is the least upper bound for a set of numbers, and it is analogous to a maximum. The companion to the least upper bound is the greatest lower bound. The least upper bound can be illustrated with a set of numbers, such as the set of numbers defined by decimal fractions beginning with a decimal point. This set has a least upper bound of 1. However, for all real numbers, the least upper bound is infinity, which is not a real number. This is because the supremum is defined as the smallest element of the set of all upper bounds, and the set of all real numbers has no upper bound. The axiom of least upper bound states that every set of real numbers that is bounded above has
  • #1
pyroknife
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The supremum is defined as the "LEAST" upper bound. The word "least" makes me think, there is a "MOST" upper bound, or at least something bigger than a "least" upper bound.

For a set of numbers, is there anything larger than a supremum? Supremum is analogous to a maximum, but I don't understand what it's called "least" upper bound.
 
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  • #2
Companion to least upper bound is greatest lower bound.

LUB is not like a maximum. It is the minimum of upper bounds.
 
  • #3
PAllen said:
Companion to least upper bound is greatest lower bound.

LUB is not like a maximum. It is the minimum of upper bounds.
Thanks, Could you illustrate this with a set of numbers? I still do not understand.
 
  • #4
pyroknife said:
Thanks, Could you illustrate this with a set of numbers? I still do not understand.
Sure. Consider the set of numbers defined by decimal fractions beginning with a decimal point. Any number greater than or equal 1 is an upper bound. The least upper bound is 1.
 
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  • #5
PAllen said:
Sure. Consider the set of numbers defined by decimal fractions beginning with a decimal point. Any number greater than or equal 1 is an upper bound. The least upper bound is 1.
Oh I think I understand. In this example, we considered only the set of numbers beginning with a decimal point.
If we consider all real numbers, is the least upper bound = upper bound = ##\infty##?
 
  • #6
pyroknife said:
Oh I think I understand. In this example, we considered only the set of numbers beginning with a decimal point.
If we consider all real numbers, is the least upper bound = upper bound = ##\infty##?
Infinity isn't a real number. So there is no LUB for all real number among the real numbers. If you consider the reals as a subset of an extended real number set, then you can make this statement true.
 
  • #7
Consider the open interval I = ]0,1[. Then, for example, that x ≤ 2, for all x ∈ I. It is also true that x ≤ 5, for all x ∈ I. This means, by definition, that 2 and 5 are upper bounds of I. A set which has (at least) one upper bound is said to be bounded above. Thus, I is bounded above, while for example [0,∞[ is not: the latter has no upper bound.
There, are, of course, infinitely many upper bounds of I. Indeed, any number u ≥ 1 is an upper bound of I. The set of all upper bounds of I is then the interval [1,∞[. This set has a smallest (least) element: s=1, which is then the least upper bound of I. This least upper bound s of I is also called the supremum of I.

The axiom of a least upper bound says that every set of real numbers which is bounded above (i.e. it has an upper bound) has a least upper bound, or a supremum. If the set is not bounded above, then it has no upper bound at all, and hence no least upper bound.
 
  • #8
For the purpose of least upper bound i.e., the sup (and greatest lower bound, i.e. the inf) of any set of numbers, it's more convenient, and standard in math, to include the possibility of either +∞ or -∞ (the plus sign is optional), if one of them is the appropriate answer. It's less convenient — and less informative — to have to say that the sup or inf is "undefined".

This way the sup and inf of an arbitrary set of real numbers is uniquely defined.

So Yes, the sup of the set of all real numbers is ∞, and its inf is -∞.

It's good to note that the sup or inf might or might not belong to that set.

(((It will probably be surprising to learn that the sup of the empty set is normally taken to be -∞, just as the inf of the empty set is normally taken to be +∞.)))
 
  • #9
The set {0, 1, 2, 3, 4} has every number larger than or equal to 4 as upper bound. The least upper bound is 4.

The set [0, 1], the set of all real numbers from 0 to 1 including both 0 and 1 has all real numbers larger than or equal to 1 as upper bounds. The least upper bound is 1.

The set (0, 1), the set of all real numbers from 0 to 1 but not including 0 and 1 still has all real numbers larger than or equal to 1 as upper bounds. The least upper bound is 1.

If an upper bound of a set is also in the set, as in the first two examples above, the upper bound is the least upper bound and is also a "maximum" for the set.

In the third example above, the set has NO maximum, or largest member, so the least upper bound is not in the set.
 

FAQ: Supremum = least upper bound, anything > supremum?

What is the definition of supremum?

The supremum of a set is the least upper bound, meaning it is the smallest number that is greater than or equal to all the numbers in the set.

What does it mean if something is greater than the supremum?

If something is greater than the supremum, it means that it is not a member of the set and is not bounded by the set's largest value.

Can a set have multiple supremums?

No, a set can only have one supremum. However, a set may not have a supremum if it is unbounded.

How is supremum different from maximum?

The supremum is the smallest number that is greater than or equal to all the numbers in a set, while the maximum is simply the largest number in the set. The maximum may or may not be included in the set, while the supremum must be included in the set.

Why is supremum important in mathematics?

Supremum is an important concept in mathematics because it allows us to define the boundaries of a set and determine its completeness. It also plays a crucial role in proving the existence of certain mathematical objects and in the foundations of calculus and analysis.

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