Results 1 to 5 of 5

Thread: least upper bound etc.

  1. #1
    Member
    Joined
    May 2008
    Posts
    94

    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.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor kalagota's Avatar
    Joined
    Oct 2007
    From
    Taguig City, Philippines
    Posts
    1,026
    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 $\displaystyle S$ is any element $\displaystyle r \in \mathbb{R}$ such that if you take any element $\displaystyle s \in S$, then $\displaystyle s \leq r$

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

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

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

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


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

    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: $\displaystyle A := \left\{ {\frac{1}{n} : n=1,2,3,...} \right\}$

    you should see that 1 is the supremum of $\displaystyle A$ and 0 is its infimum.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    10
    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 $\displaystyle (A, \leq)$ be a partially ordered set. Let $\displaystyle X\subset A$ be a non-empty subset. We say $\displaystyle a\in A$ is an upper bound for $\displaystyle X$ if $\displaystyle x\leq a$ for all $\displaystyle x\in X$. Similarly we say $\displaystyle b\in A$ is a lower bound for $\displaystyle X$ if $\displaystyle b\leq x$ for all $\displaystyle x\in X$. Note we do not require that $\displaystyle a,b\in X$, they do not have to. We define the least upper bound to be $\displaystyle a\in A$, an upper bound, so that if $\displaystyle x\leq b$ for all $\displaystyle x\in A$ then $\displaystyle a\leq b$, hence the name "upper bound". Look at what kalagota did, though we was working with $\displaystyle \mathbb{R}$ rather than any arbitrary set the concepts still apply. Use his examples to help motivate the more general abstract definition.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Member
    Joined
    May 2008
    Posts
    94
    Quote Originally Posted by ThePerfectHacker View Post
    Let $\displaystyle (A, \leq)$ be a partially ordered set. Let $\displaystyle X\subset A$ be a non-empty subset. We say $\displaystyle a\in A$ is an upper bound for $\displaystyle X$ if $\displaystyle x\leq a$ for all $\displaystyle x\in X$. Similarly we say $\displaystyle b\in A$ is a lower bound for $\displaystyle X$ if $\displaystyle b\leq x$ for all $\displaystyle x\in X$. Note we do not require that $\displaystyle a,b\in X$, they do not have to. We define the least upper bound to be $\displaystyle a\in A$, an upper bound, so that if $\displaystyle x\leq b$ for all $\displaystyle x\in A$ then $\displaystyle a\leq b$, hence the name "upper bound". Look at what kalagota did, though we was working with $\displaystyle \mathbb{R}$ 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?
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor kalagota's Avatar
    Joined
    Oct 2007
    From
    Taguig City, Philippines
    Posts
    1,026
    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..
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 0
    Last Post: Feb 19th 2010, 01:06 AM
  2. greatest least bound and least upper bound proof
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: Nov 4th 2009, 04:44 PM
  3. Upper bound/Lower bound?
    Posted in the Pre-Calculus Forum
    Replies: 7
    Last Post: Sep 13th 2009, 10:48 AM
  4. upper bound
    Posted in the Calculus Forum
    Replies: 3
    Last Post: Jul 4th 2008, 09:42 AM
  5. least upper bound and greatest lower bound
    Posted in the Calculus Forum
    Replies: 2
    Last Post: Sep 22nd 2007, 09:59 AM

Search Tags


/mathhelpforum @mathhelpforum