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Is this statement true , for all numbers " If A * B * C = X * Y * Z , THEN {X,Y,Z}={A,B,C} " ? If not , when is this statement true ? I ask this question as I have came across the solution of a system of equations in a textbook , it concludes with As (Z-x)(Z-y)(Z-z) = (Z-a)(Z-b)(Z-c) , then (x,y,z) is a permuation on {a,b,c} . ( Z (capital Z) is a a variable in the set of complex numbers, we solve for (x,y,z) )

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    The first statement you make is clearly not correct. $1\times 1\times 8=2\times 2\times 2$. Of course, a monic polynomial over $\mathbb C$ determines and is determined by its roots. – lulu Jun 22 '20 at 14:55
  • It is vitally important to be clear about which quantifiers control which parts of a statement. “For all ... if” is very different from “if for all ... “. – David K Jun 22 '20 at 15:00
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    The first statement is not true, as I'm sure you suspect. For example, $2\cdot 8\cdot 6 = 3\cdot 4\cdot 8$ but the set ${2,8,6}$ is not the same as ${3,4,8}$, is it? However when you introduce the variable $Z$ in the second statement, this provides much stronger information, an equality of polynomials, where all the corresponding coefficients have to be equal. The second statement is valid. – hardmath Jun 22 '20 at 15:00
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    By Fundamental theorem of algebra the statement from the textbook is true. It is not equivalent to the first one, however. – user Jun 22 '20 at 15:21

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