basic algebra

Discussion in 'Undergraduate Math' started by Chergarj, Jun 23, 2003.

  1. Chergarj

    Chergarj Guest

    if x<y and z<q can I assert that x+z<y+q?
    No.

    My answer is based on intuition, and sometimes intuition by itself could be
    wrong.

    G C
     
    Chergarj, Jun 23, 2003
    #1
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  2. Yes.

    A quickie proof
    x<y
    x - y < y - y
    x - y < 0

    z<q
    z - z < q - z
    0 < q - z

    Since x - y < 0 and 0 < q - z, then x - y < q - z
    x - y < q - z
    x - y + z < q - z + z
    x - y + z < q
    x - y + z + y < q + y
    x + z < q + y
    x + z < y + q QED
     
    Rich Carreiro, Jun 23, 2003
    #2
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  3. Chergarj

    Vigual Guest

    if x<y and z<q can I assert that x+z<y+q?
     
    Vigual, Jun 23, 2003
    #3
  4. Chergarj

    George Cox Guest


    It depends on what rules govern "<" and "+". In a ring a set P of
    so-called positive elements has the property that x and y in P implies x
    + y in P, and x <y is defined to mean y - x in P. So you have

    y - x in P and q - z in P,
    hence

    (y - x) + (q - z) in P

    Now ring manipulations give

    (y + q) - (x + z) in P.

    Though rings are the usual setting for such things, you can do the same
    in an additive group. But can it be done in _your_ system? I don't
    know.

    GC
     
    George Cox, Jun 23, 2003
    #4
  5. Chergarj

    Stan Brown Guest

    Yes, and what is more, your assertion will be correct. :)
     
    Stan Brown, Jun 24, 2003
    #5
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