The discussion revolves around the concept of "supremum" in mathematics, particularly in the context of inequalities involving variables x and y. Participants explore the definition of supremum as the "least upper bound" and its implications for different sets of numbers, including cases where x and y are treated as free variables or specific values. The scope includes theoretical understanding and clarification of mathematical definitions.
Participants do not reach a consensus on the definitions and implications of supremum in this context. Multiple competing views remain regarding the treatment of x and y as variables or fixed values, and the existence of supremum for different sets.
Limitations include the ambiguity in defining the sets for x and y, and the dependence on whether x and y are treated as free variables or specific values. The discussion also reflects varying interpretations of mathematical definitions related to supremum.
The least upper bound of what? x or y? If both are free variables then it is rather useless. A "least upper bound" is the smallest number that is greater than or equal to every number in a set. In most of the interesting cases the set is infinite.pyroknife said:If let's say I have an expression:
##x\leq y##
Since the supremum is defined as the "least upper bound," does this make sup(x) for this case ##x=y## or is it ##x = \infty##?
In this case I am asking for the supremum of x, sup(x), which would be the least upper bound of x. Would it be x = y?Hornbein said:The least upper bound of what? x or y? If both are free variables then it is rather useless. A "least upper bound" is the smallest number that is greater than or equal to every number in a set. In most of the interesting cases the set is infinite.
Maybe fixed after you commented, but OP reads sup(x). Then, sup(x)=y is correct for the set of x ≤ y, assuming y is a given value. Of course, this is a totally trivial case.Hornbein said:The least upper bound of what? x or y? If both are free variables then it is rather useless. A "least upper bound" is the smallest number that is greater than or equal to every number in a set. In most of the interesting cases the set is infinite.
You wouldn't say x=y, you would say sup(x)=y, for the specified set.pyroknife said:In this case I am asking for the supremum of x, sup(x), which would be the least upper bound of x. Would it be x = y?
Thanks. If instead we want sup(y), would that be infinity?PAllen said:Maybe fixed after you commented, but OP reads sup(x). Then, sup(x)=y is correct for the set of x ≤ y, assuming y is a given value. Of course, this is a totally trivial case.
ah yes, thanksPAllen said:You wouldn't say x=y, you would say sup(x)=y, for the specified set.
But as you wrote it, it seems y is just one number, and not a set, so, trivially, sup{y}=y (on most "reasonable" choices of orderings , where ## a \leq a ##)pyroknife said:Thanks. If instead we want sup(y), would that be infinity?
I am asking this question as a follow up of another thread I made right below this one regarding norms and supremums.
That depends on how you define the set against which x≤y is defined. That is, given x, and the set of y such that x ≤ y of real numbers, then sup(y) does not exist. If, instead, you use the set of extended reals that includes +/- ∞ as wellas the conventional reals, then, indeed, sup(y)=∞.pyroknife said:Thanks. If instead we want sup(y), would that be infinity?
I am asking this question as a follow up of another thread I made right below this one regarding norms and supremums.
PAllen said:That depends on how you define the set against which x≤y is defined. That is, given x, and the set of y such that x ≤ y of real numbers, then sup(y) does not exist. If, instead, you use the set of extended reals that includes +/- ∞ as wellas the conventional reals, then, indeed, sup(y)=∞.
WWGD said:But as you wrote it, it seems y is just one number, and not a set, so, trivially, sup{y}=y (on most "reasonable" choices of orderings , where ## a \leq a ##)