Results 1 to 4 of 4
Like Tree2Thanks
  • 1 Post By SlipEternal
  • 1 Post By SlipEternal

Math Help - Prove that if a|b and b|a, then a=b or a=-b

  1. #1
    Junior Member
    Joined
    Sep 2013
    From
    United States
    Posts
    59
    Thanks
    1

    Prove that if a|b and b|a, then a=b or a=-b

    Course: Foundations of Higher Math

    This is in the chapter titled, "More on Direct Proof and Proof by Contrapositive"

    Let a,b\in Z, where a\neq 0 and b\neq 0. Prove that if a|b and b|a , then a=b or a=-b

    Proof by contrapositive seems too difficult, so I'm trying a direct proof.

    Assume that a|b and b|a, i.e. b=ax and a=by, for some x,y\in Z

    Then,

    a=b \Rightarrow by=ax \Rightarrow (ax)y=(by)x \Rightarrow a(xy)=b(yx). Then divide both sides by (xy), so a=b

    Is this sufficient?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Nov 2010
    Posts
    1,407
    Thanks
    523

    Re: Prove that if a|b and b|a, then a=b or a=-b

    Again, you are starting with the conclusion when you write a = b \Rightarrow \cdots. Starting with the conclusion is circular. When you arrive back at the same conclusion, it is based on the fact that you assumed it to start with.

    You were correct to assume a|b and b|a. So begin with the equalities you have from those assumptions.

    b = ax, a = by

    Start with one of them: b = ax \Rightarrow b = (by)x By cancellation, 1 = xy. The only integers whose product is 1 are x=y=1 or x=y=-1. So, a = b(1) or a = b(-1).
    Thanks from MadSoulz
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Sep 2013
    From
    United States
    Posts
    59
    Thanks
    1

    Re: Prove that if a|b and b|a, then a=b or a=-b

    Ok. Take a look at this:

    b = ax \Rightarrow b = (by)x \Rightarrow b=b(xy) \Rightarrow b - b(xy) = 0 The only way that b-b(xy)=0 is when xy = 1. The only way that xy =1 is if x=y=1, as you said. So a=b(1)

    I can't work it out get a=b(-1) though.
    Last edited by MadSoulz; October 8th 2013 at 05:52 PM.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor
    Joined
    Nov 2010
    Posts
    1,407
    Thanks
    523

    Re: Prove that if a|b and b|a, then a=b or a=-b

    There are two ways for a product of two integers to be 1. Either x=y=1 OR x=y=-1. Your professor may ask you to prove that those are the only possibilities, which is simple. Just show that if x and y have opposite signs, then their product is negative. Hence, the have the same sign. Suppose they are both positive and x>1 then xy \ge x>1. Next, suppose they are both negative and x<-1 then xy \ge x(-1) = -x > 1.
    Thanks from MadSoulz
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Prove that if A ~ B then P(A) ~ P(B).
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: November 29th 2010, 09:36 AM
  2. Replies: 2
    Last Post: August 28th 2009, 02:59 AM
  3. prove
    Posted in the Number Theory Forum
    Replies: 2
    Last Post: September 18th 2008, 05:22 AM
  4. Prove
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: September 17th 2008, 12:57 PM
  5. prove that
    Posted in the Algebra Forum
    Replies: 4
    Last Post: September 7th 2008, 05:14 PM

Search Tags


/mathhelpforum @mathhelpforum