Results 1 to 4 of 4

Math Help - order of reals

  1. #1
    Newbie
    Joined
    Sep 2008
    Posts
    4

    order of reals

    1. Prove that < is an order for for all real numbers.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    is up to his old tricks again! Jhevon's Avatar
    Joined
    Feb 2007
    From
    New York, USA
    Posts
    11,663
    Thanks
    3
    Quote Originally Posted by princess08 View Post
    1. Prove that < is an order for for all real numbers.
    it may help if you tell us what kind of axioms you are using and how your text/class/professor defines "order"
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Sep 2008
    Posts
    4
    Defn.
    From the introductory lectures, an ordered set is a set S with a relation < which satisfies
    two properties:
    1. (Trichotomy property) for any two elements
    a, b is an element of S exactly one of the following hold a < b, a = b, or b < a.

    2. (Transitive property) for any three elements
    a, b, c is an element ofS, if a < b and b < c, then a < c.

    In this case the relation
    < is called an order.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    is up to his old tricks again! Jhevon's Avatar
    Joined
    Feb 2007
    From
    New York, USA
    Posts
    11,663
    Thanks
    3
    Quote Originally Posted by princess08 View Post
    Defn.
    From the introductory lectures, an ordered set is a set S with a relation < which satisfies
    two properties:
    1. (Trichotomy property) for any two elements
    a, b is an element of S exactly one of the following hold a < b, a = b, or b < a.

    2. (Transitive property) for any three elements
    a, b, c is an element ofS, if a < b and b < c, then a < c.

    In this case the relation
    < is called an order.
    ok, so you want to prove these two things for the relation "<" on the real numbers. (by the way, if it has to work for all real numbers, shouldn't we be considering " \le", also, what do you know about real numbers? are we allowed to talk about "negative" and "positive" here? the proof here is subtle, since " \le" is defined on the real numbers to fit these conditions, so it might seem like begging the question if we use concepts that we aren't allowed to)

    so, (1) Let a,b \in \mathbb{R}. Is it true that a = b, or a < b or b < a? how would you show that? does considering (a - b) help?

    and (2) Let a,b,c \in \mathbb{R}, and assume that a < b and b < c according to the relation of "<" on the reals. then that would mean that a - b < 0 and b - c < 0. what about their sum? (the reals are closed under addition, thus we can use the idea of "sum")
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 1
    Last Post: August 23rd 2010, 08:23 AM
  2. Replies: 1
    Last Post: May 14th 2010, 01:53 AM
  3. Measuring the reals
    Posted in the Differential Geometry Forum
    Replies: 4
    Last Post: December 18th 2009, 06:02 AM
  4. All reals numbers
    Posted in the Algebra Forum
    Replies: 2
    Last Post: October 19th 2009, 10:43 PM
  5. Topology of the Reals
    Posted in the Calculus Forum
    Replies: 1
    Last Post: February 9th 2009, 05:09 PM

Search Tags


/mathhelpforum @mathhelpforum