Relating an expression to two similar onesIs it possible to express C solely in terms of A and B, where A = m x + z , B = n y + z , C = m + n x + y + z and m , n , x , y , z &gt; 0 ?If not, how close can I get?

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Relating an expression to two similar onesIs it possible to express C solely in terms of A and B, where  A  =            m              x        +        z              ,  B  =            n              y        +        z              ,  C  =                    m        +        n                    x        +        y        +        z            and   m  ,  n  ,  x  ,  y  ,  z  &amp;gt;  0     ?If not, how close can I get?

Cordell Crosby · Accepted Answer

Here is a proof by counterexample that you cannot express C solely in terms of A and B:Let&#039;s take   a  =      1    6   and    B  =      1    8    and get these values in two ways:  m  =  n  =  1  ,   (  x  ,  y  ,  z  )  =  (  1  ,  5  ,  3  )  and      m    ′    =  1  ,       n    ′    =      8    7    ,   (      x    ′    ,      y    ′    ,      z    ′    )  =  (  2  ,  4  ,  3  )Then using   m  ,  n  ,  x  ,  y  ,  z we get       C    ′    =      5    21  . There are two different values of C for identical values of A and B.Let&#039;s phrase your second question more precisely:Let       C    max    (  A  ,  B  )  ≡      max          m      ,      n      ,      x      ,      y      ,      z                  m      +      n              x      +      y      +      z        :      (          m              x        +        z              =    A    )    ∧      $        (          n              y        +        z              =    B    )  and      C    max    (  A  ,  B  )  ≡      min          m      ,      n      ,      x      ,      y      ,      z                  m      +      n              x      +      y      +      z        :      (          m              x        +        z              =    A    )    ∧      $        (          n              y        +        z              =    B    )  Find       C    max    (  A  ,  B  )  −      C    min    (  A  ,  B  ) as a function of A and B.Unfortunately, you can&#039;t get close at all to the value of C in terms of A and B because for fixed A and B (unless A=B=0),  C  =  A  +  (  B  −  A  )      y          x      +      y      +      z        +  B      z          x      +      y      +      z      so choosing  x  =  −  1  −  (  k  −  1  )  ϵ    y  =      1    2    +  ϵ    z  =      1    2    +  k  ϵand selecting m,n to give the correct values of A,B,both    (  B  −  A  )      y          x      +      y      +      z       and   B      z          x      +      y      +      z       blow up as   ϵ goes to zero, if the corresponding numerator is non-zero. For any particular values of (A,B) other than (0,0), we can choose k such that those two terms are not negatives of one another. So we can make

Relating an expression to two similar ones Is it possible to express C solely in terms of A and B,

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