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2bornot2b Group Title

What is the significance of the phrase "non empty"?

  • 2 years ago
  • 2 years ago

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  1. 14yamaka Group Title
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    It means that a set does have elements, and is not an empty set.

    • 2 years ago
  2. 2bornot2b Group Title
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    I find the phrase "nonempty" absent in the definition of supremum, but it is present in the definition of completeness axiom. Can you explain why?

    • 2 years ago
  3. 2bornot2b Group Title
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    Should I write down the definitions here?

    • 2 years ago
  4. JamesJ Group Title
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    I know what they are. I'm sure in the context, it's clear that the set for which you're finding the sup is not empty. If it were, the sup wouldn't be defined.

    • 2 years ago
  5. 2bornot2b Group Title
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    Is it " If it were" or " If it weren't"

    • 2 years ago
  6. 2bornot2b Group Title
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    I didn't get you ...

    • 2 years ago
  7. JamesJ Group Title
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    If a subset of the real number, \( A \subset \mathbb{R} \), were empty, then it would have no sup.

    • 2 years ago
  8. JamesJ Group Title
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    Hence \[ \sup A \] is defined if and only if \( \emptyset\neq A \subset \mathbb{R} \) and \( A \) has an upper bound.

    • 2 years ago
  9. JamesJ Group Title
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    the least upper bound is the supremum. Different words, same thing. Likewise for glb and inf

    • 2 years ago
  10. JamesJ Group Title
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    I am. The lub = least upper bound. The lub is the supremum. Same thing.

    • 2 years ago
  11. 2bornot2b Group Title
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    And why is it absent in the definition of upperbound

    • 2 years ago
  12. JamesJ Group Title
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    upper bound or least upper bound?

    • 2 years ago
  13. 2bornot2b Group Title
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    Just upper bound

    • 2 years ago
  14. JamesJ Group Title
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    Give me the definition you're talking about. But in any case, we say the empty set has no upper or lower bound; we just don't define it if the set is empty.

    • 2 years ago
  15. 2bornot2b Group Title
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    Definition of Upper Bound: Let S be a set of real numbers. If there is a real number b such that \[x\le b\] for every x in S, then b is called an upper bound for S and we say that S is bounded above by b

    • 2 years ago
  16. 2bornot2b Group Title
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    My point is why not say "let S be a non empty set of real numbers"

    • 2 years ago
  17. JamesJ Group Title
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    Very technically, you are correct. Because by this definition, every real number is an upper bound of the empty set. And that leads to a clash with the completeness axiom because there is no least upper bound, no sup of the empty set. So to avoid this problem, we should say the empty set has no upper bound. agreed.

    • 2 years ago
  18. 2bornot2b Group Title
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    But I copied that definition from T.M. Apostol, of course someone before me would have pointed this out, if it were not acceptable.

    • 2 years ago
  19. JamesJ Group Title
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    It could also be clear in the context in which this definition is given that the subset is non-empty. But in any case, whatever this book is, it isn't the word of god. Take it from me that, yes: this only makes sense if the subset is non-empty. And move on.

    • 2 years ago
  20. 2bornot2b Group Title
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    Sorry my computer crashed, and it got restarted.

    • 2 years ago
  21. 2bornot2b Group Title
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    Also take a look at this definition of neighbourhood of a point (Its from the lecture note of our prof) Let \[c\in \mathbb{R}\]. A subset S of R is said to be a neighbourhood of c, if there exists an open interval (a,b) such that \[c\in (a,b)\subset S\]

    • 2 years ago
  22. 2bornot2b Group Title
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    I mean I would like to state it as "Let c∈R. A non empty subset S of R is said to be a neighbourhood of c, if there exists an open interval (a,b) such that c∈(a,b)⊂S"

    • 2 years ago
  23. JamesJ Group Title
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    An open interval is by definition necessarily non-empty, so in this case, I would argue it's not necessary. To be more explicit, an open interval is a subset \[ T = \{ x \ | \ a < x < b \} \] where \( a < b \) and therefore \( T \) is not empty.

    • 2 years ago
  24. 2bornot2b Group Title
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    Have a look at this http://en.wikipedia.org/wiki/Empty_set#Extended_real_numbers The first line.. that starts with "since the empty set........."

    • 2 years ago
  25. JamesJ Group Title
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    Right, whatever that article is talking about "Extended Reals" or whatever else you want to call it, that's not the Real Numbers as well usually define them.

    • 2 years ago
  26. 2bornot2b Group Title
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    Okay,... I am getting you.... Thanks.

    • 2 years ago
  27. JamesJ Group Title
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    **we [not well]

    • 2 years ago
  28. 2bornot2b Group Title
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    what did you mean by that, I am sorry I didn't understand that...

    • 2 years ago
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