Results 1 to 2 of 2

Math Help - How would I write a proof for these partial orders?

  1. #1
    Newbie
    Joined
    Dec 2013
    From
    United States
    Posts
    1

    How would I write a proof for these partial orders?

    Let R 1 and R 2 be relations on N defined by

    xR 1 y if and only if y=a+x for some a∈N 0 .

    xR 2 y if and only if y=xa for some a∈N

    For all x;y∈N.

    Also N 0 denotes all integers x≥0 , while N denotes all integers x≥1 .

    There are two different things I want to prove with this.
    1.
    I want to write a proof to show that R 1 is a partial order on N .

    2.
    I want to write a proof to show that R 2 is a partial order on N .


    I have others that I want to try and do but for now if someone could model how to write a proof for these that would be great as I could reference to it.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Nov 2013
    From
    California
    Posts
    2,645
    Thanks
    1060

    Re: How would I write a proof for these partial orders?

    Quote Originally Posted by mj2323 View Post
    Let R 1 and R 2 be relations on N defined by

    xR 1 y if and only if y=a+x for some a∈N 0 .

    xR 2 y if and only if y=xa for some a∈N

    For all x;y∈N.

    Also N 0 denotes all integers x≥0 , while N denotes all integers x≥1 .

    There are two different things I want to prove with this.
    1.
    I want to write a proof to show that R 1 is a partial order on N .

    2.
    I want to write a proof to show that R 2 is a partial order on N .


    I have others that I want to try and do but for now if someone could model how to write a proof for these that would be great as I could reference to it.
    I see that A relation R on S that is reflexive, anti-symmetric, and transitive is said to be a partial order on S.

    So you need to show each of these for your relation.

    reflexive means (x R x)

    antisymmetric means if (x R y) and (y R x) then x = y

    transitive means if (x R y) and (y R z) then (x R z)

    Ok so now show each of these for your relations.

    R1: (x R1 y) iff y=a+x for a a non-negative integer.

    does (x R1 x)? x = a+x --> a = 0, and 0 is a non-negative integer so yes R1 is reflexive

    suppose (x R1 y), then y = a+x, if (y R1 x) then x = a+y, so x = a + a + x, a = 0, and thus x = y. So R1 is anti-symmetric

    suppose (x R1 y), and (y R1 z). then y = a1+x, z = a2 + y, so z = (a1+a2) + x, (a1+a2) is in N0, and so (x R1 z), and R1 is transitive.

    Now you do R2.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Partial Orders
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: February 12th 2013, 04:40 AM
  2. Partial orders
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: May 6th 2010, 12:34 AM
  3. Partial Orders
    Posted in the Discrete Math Forum
    Replies: 0
    Last Post: October 15th 2009, 03:20 PM
  4. partial orders
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: October 1st 2009, 06:46 AM
  5. Partial Orders
    Posted in the Discrete Math Forum
    Replies: 5
    Last Post: May 2nd 2009, 08:28 PM

Search Tags


/mathhelpforum @mathhelpforum