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Discrete Mathematics Absolute Value Proof

  1. Sep 12, 2007 #1
    1. The problem statement, all variables and given/known data
    Prove the following statement:
    For all real numbers x and y, |x| times |y| = |xy|



    2. Relevant equations
    I really don't know how to start this as a formal proof.


    3. The attempt at a solution
    I was thinking I'd have to break it down into four cases and logically prove that the statement is true because no matter what, x times y is going to have the same numerical value as it's opposite number (of course beside it being negative) because once you take the absolute value, it's going to be positive anyways.
    Case 1: Suppose both x and y are positive real numbers.
    Case 2: Suppose x is a negative real number and y is a positive real number.
    Case 3: Suppose x is a positive real number and y is a negative real number.
    Case 4: Suppose both x and y are positive.

    Am I on the right track or am I going in the wrong direction?
     
    Last edited: Sep 12, 2007
  2. jcsd
  3. Sep 12, 2007 #2
    if you want to do cases you only need to do 3. You can WLOG two of them together.
     
  4. Sep 12, 2007 #3
    Do you mean cases 2 and 3 then?
     
  5. Sep 12, 2007 #4

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    Yes, the situation where a> 0 and b< 0 is exactly the same as a< 0 and b>0. However, I would not discourage you from considering the two cases separately. You are completely correct to argue that there are 2 cases for x and 2 cases for y and so (2)(2)= 4 cases altogether.
     
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