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Unbounded sets

  1. Mar 4, 2010 #1
    Is a bounded set synonymous to a set that goes to infinity? I feel like unless a set is
    (-infinity, n) or [n, infinity) it is not going to be unbounded.

    The other thing that I was wondering is can a set be neither open nor closed AND unbounded? Doesn't the definition of open/closed imply that there is a boundary?

  2. jcsd
  3. Mar 5, 2010 #2
    A set being bounded just means it is contained in some ball. Boundedness isn't really related to being open or closed, and also has nothing to do with boundary.
  4. Mar 5, 2010 #3
    No and furthermore the bound does not even have to be a member of the set.

    A set can have lots of bounds, even an infinite number of them.

    A bound can be 'above' or 'below'.

    An upper bound is simply a real number that is either greater than or equal to every member of the set or less than/equalto every member of the set.

    so 7,8,9,10 etc all form upper bounds to the set {2,3,4,5,6} ; none are memebrs of the set.
    but wait 6 also forms an upper bound as it satisfies the equal to and is a member.

    Similarly 1,0,-1,-2 all form lower bounds that are not members and 2 forms a lower bound that is

    Sets which 'go to infinity' are unbounded. However a set can contain an infinite number of members and still be bounded, above and/or below.

    for example the set {1/1, 1/2, 1/3, .....} is bounded above by 2 ,1 etc and bounded below by 0, -1 etc, but contains an infinite number of members.

    The set [tex]\{ - \infty ,..... - 2, - 1,0,1,\frac{1}{2},\frac{1}{3},.......\} [/tex]

    is not bounded below but is bounded above.

    If a set has both a lower and upper bound so that the modulus of any member, x, is less than or equal to some real number K then the set is bounded. (No upper or lower)

    If for any [tex]x \in S[/tex] there exists a K such that

    [tex]\left| x \right| \le K[/tex]

    The set S is bounded.

    Hope this helps.
  5. Mar 5, 2010 #4
    "bounded" does not mean "has a boundary"

    Isn't English confusing?
  6. Mar 5, 2010 #5
    So I guess what I'm really wondering is can a set be unbounded if it doesn't go to infinity? That's what I can't seem to wrap my brain around, because I feel that if a set doesn't go to infinity, there will always be a real number K larger than the members of the set.
  7. Mar 5, 2010 #6
    Any finite set S of real numbers is bounded.
    Infinite sets can be bounded or unbounded.

    Remember that to be bounded a set must have both a lower bound and an upper bound.
    Sets with only a lower or upper bound are unbounded.

    A further question for you to ponder:

    Can a set of complex numbers be bounded?
    Last edited: Mar 5, 2010
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