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Math Help - least upper bound etc.

  1. #1
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    least upper bound etc.

    Can anyone give me a good explanation of...

    a) upper bound
    b) lower bound
    c) least upper bound
    d) greatest lower bound

    It's a chapter of Relations - "Partial ordering". My text doesn't give a good explanation about it.
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  2. #2
    MHF Contributor kalagota's Avatar
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    Quote Originally Posted by robocop_911 View Post
    Can anyone give me a good explanation of...

    a) upper bound
    b) lower bound
    c) least upper bound
    d) greatest lower bound

    It's a chapter of Relations - "Partial ordering". My text doesn't give a good explanation about it.


    a. an upper bound of the set S is any element r \in \mathbb{R} such that if you take any element s \in S, then s \leq r

    b. a lower bound of the set S is any element r \in \mathbb{R} such that if you take any element s \in S, then r \leq s

    note that upper bounds and lower bounds are not necessarily in S.

    c. the least upper bound of S, or we call the supremum of S \sup S is an upper bound u of S such that if you take any upper bound u_0 of S, u \leq u_0. In other words, \sup S is the smallest among all the upper bounds.

    d. the greatest lower bound of S, or we call the infimum of S \inf S is a lower bound v of S such that if you take any lower bound v_0 of S, v_0 \leq v. In other words, \inf S is the biggest (greatest) among all the lower bounds.


    Example:
    Consider the set \{1, 2, 3, 4, 5\}

    these are some upper bounds.. 5, 5.1, 6, 7, 6.5,.. in fact, any number greater than or equal to 5 is an upper bound. While the supremum of the set if 5 since if you take all the upper bounds, 5 is the smallest.. Hope you can tell what the lower bounds are, and what the infimum is.

    another example: A := \left\{ {\frac{1}{n} : n=1,2,3,...} \right\}

    you should see that 1 is the supremum of A and 0 is its infimum.
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  3. #3
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    Quote Originally Posted by robocop_911 View Post
    Can anyone give me a good explanation of...

    a) upper bound
    b) lower bound
    c) least upper bound
    d) greatest lower bound

    It's a chapter of Relations - "Partial ordering". My text doesn't give a good explanation about it.
    Let (A, \leq) be a partially ordered set. Let X\subset A be a non-empty subset. We say a\in A is an upper bound for X if x\leq a for all x\in X. Similarly we say b\in A is a lower bound for X if b\leq x for all x\in X. Note we do not require that a,b\in X, they do not have to. We define the least upper bound to be a\in A, an upper bound, so that if x\leq b for all x\in A then a\leq b, hence the name "upper bound". Look at what kalagota did, though we was working with \mathbb{R} rather than any arbitrary set the concepts still apply. Use his examples to help motivate the more general abstract definition.
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    Quote Originally Posted by ThePerfectHacker View Post
    Let (A, \leq) be a partially ordered set. Let X\subset A be a non-empty subset. We say a\in A is an upper bound for X if x\leq a for all x\in X. Similarly we say b\in A is a lower bound for X if b\leq x for all x\in X. Note we do not require that a,b\in X, they do not have to. We define the least upper bound to be a\in A, an upper bound, so that if x\leq b for all x\in A then a\leq b, hence the name "upper bound". Look at what kalagota did, though we was working with \mathbb{R} rather than any arbitrary set the concepts still apply. Use his examples to help motivate the more general abstract definition.
    How do we find out if x<= a or x>=a if the set consists of {a,b,c,d...}.

    Should it be lexicographic?
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  5. #5
    MHF Contributor kalagota's Avatar
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    Quote Originally Posted by robocop_911 View Post
    How do we find out if x<= a or x>=a if the set consists of {a,b,c,d...}.

    Should it be lexicographic?
    i think so.. most of the time, the alphabet is mapped to the set of natural numbers, i.e., 1 for a, 2 for b, etc.. but also consider what your teacher told you about the mapping..
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