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Supremum and infimum of a set

  1. Aug 30, 2006 #1
    let [tex] B = \{x\in\mathbb{R} : sinx \geq 0 \} [/tex]

    find the supremum and infimum of this set.

    Ok well, since it is periodic I guess the point would be to note that the set will repeat ever [tex]2\pi[/tex]

    So then if we consider just between 0 and [tex]2\pi[/tex]

    supremum = [tex]\pi[/tex]
    infimum = 0

    if we consider all [tex]\mathbb{R}[/tex]

    here is where I'm confused. The supremum would just be the [tex]N\pi[/tex] when N is an odd integer. Should I just state the function is periodic it will repeat between 0 and [tex]2\pi[/tex]
    Last edited: Aug 30, 2006
  2. jcsd
  3. Aug 30, 2006 #2


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    Proceed methodically from the definition. M is a supremum of B if it is the smallest superior bound. But is B even bounded superiorly?
  4. Aug 30, 2006 #3

    Ok so I think I see what your saying. The set will not be bounded above or below except by plus or minus infinity b/c the function is periodic. I can always find a larger number in the reals that satifies sin(x) greater than or equal to 0. Therefore the set would have a supremum or positive infinity and a infimum of negative infinity.

    Oh, I made typo in the original problem [tex]x\in\mathbb{R}_e[/tex]
  5. Aug 30, 2006 #4


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    That is the idea, yeah. You'd have to write a few equations though for it to be considered a proof. You'd have to show rigorously that given any number in B, there is always another number in B that is superior(resp. inferior) to it.

    (What is [itex]\mathbb{R}_e[/itex]??)
  6. Aug 30, 2006 #5
    Our professor stated that [tex]\mathbb{R}_e[/tex] is the extended reals which contains plus and minus infinity. This course is an analysis for electrical engineers we get a crash course in a little bit of set theory then a bunch about complex functions with linear algebra of complex functions. We don't have a textbook for this course and the professors only written some of the course notes so I'm kind of flying blind on what is going on here. Thanks for the help!
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