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Is the following statement true or false. prove?

  1. Apr 26, 2012 #1
    1. The problem statement, all variables and given/known data

    let a,b,c be three integers is a divides c and b divides c, then either a divides b or b divides a.


    if a,b,c,d are real numbers with a<b and c<d, then ac<bd.



    2. Relevant equations



    3. The attempt at a solution

    So for part 1

    Since a/c and b/c

    then, c=ak and c=bd for some integers k,d.

    hence, ak=bd

    so a/b=d/k

    where d/k =l for some integer l.

    so a/b=l... Therefore a/b. end of proof.

    for part 2)


    a<b and c<d (given).

    let a= -5 and b=-4 satisfying a<b

    and let c=-7 and d=-5 satisfying c<d

    so then ac

    = 35

    and bd

    = 20

    so 35<20

    but this is false

    therefore ac<bd is false. end of proof.

    Can Someone please tell me if I have done this correctly?
     
  2. jcsd
  3. Apr 26, 2012 #2
    How do you know d/k is an integer? Try a few values out for a, b, and c and see how the corresponding d and k correlate. You should see the answer to the problem pretty quickly!

    The second counterexample looks good!
     
  4. Apr 26, 2012 #3
    So for the first part can I just use examples and then prove it false I guess I can, but I would like to know how to prove it algebraically without having to use real number, since most of the proofs I am doing at the moment involve this.

    Ok so

    a/c and b/c (given)

    let c=3 and a=9 and b=12

    so 9=3k for some positive integer k

    and 12=3s for some positive integer s

    but 12 does not divide 9 and 9 does not divide 12. therefore a/b or b/a is false.

    but obviously I can make this statement true ie if k=6 and s=12... Although I guess this in not a "for some" statement... but I really don't like doing proofs this way.
     
  5. Apr 26, 2012 #4
    Well, if a statement is sfalse, the only way to really prove it is to provide a counterexample.

    In your example, 9 and 12 don't both divide 3. You have the definition backwards, a and b will be multiplied by the interger terms.
     
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