This simple looking problem is making me crazy! Plz help!

[bhaghpan]

Assume x1, x2, x3, …, xn are real numbers. Let A be the following set of n equations:

x1>0 ,
x2>0 ,
x3>0 ,
…
xn>0 .

And let B be the following set of n equations:

x1 + x2 + … + xn >0 ,
x1*x2 + x1*x3 + … + x1*xn + x2*x3 + x2*x4 + … + xn-1*xn >0 or (∑xi*xj>0 , (i≠j)) ,
x1*x2*x3 + … >0 or (∑xi*xj*xk>0 , (i≠j , i≠k , k≠j)) ,
…
x1*x2*x3*…*xn >0 .

Show A is true if and only if B is true.

 

[bhaghpan]

Assume x1, x2, x3, …, xn are real numbers. Let A be the following set of n equations:

x1>0 ,
x2>0 ,
x3>0 ,
…
xn>0 .

And let B be the following set of n equations:

x1 + x2 + … + xn >0 ,
x1*x2 + x1*x3 + … + x1*xn + x2*x3 + x2*x4 + … + xn-1*xn >0 or (∑xi*xj>0 , (i≠j)) ,
x1*x2*x3 + … >0 or (∑xi*xj*xk>0 , (i≠j , i≠k , k≠j)) ,
…
x1*x2*x3*…*xn >0 .

Show A is true if and only if B is true.

 

[stapel]

Assume x₁, x₂, x₃, …, x_n are real numbers. Let A be the following set of n equations:

x₁ > 0 ,
x₂ > 0 ,
x₃ > 0 ,
…
x_n > 0 .

And let B be the following set of n equations:

x₁ + x₂ + … + x_n > 0 ,

x ₁ *x ₂ + x ₁ *x ₃ + … + x ₁ *x _n + x ₂ *x ₃ + x ₂ *x ₄ + … + x _{n -1}* x_n > 0 (or ∑x_i*x_j > 0 , (i≠j)) ,

x₁*x₂*x₃ + … > 0 (or ∑x_i*x_j*x_k > 0 , (i≠j , i≠k , k≠j)) ,

…

x₁*x₂*x₃*…*x_n >0 .

Show A is true if and only if B is true.

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By "A is true", do you mean "each of the equations contained in A is true"? (Same questions for "B is true".)

What results (algorithms, techniques, rules, theorems, etc) from your class do you think might usefully be applied to this proof? What have you tried? How far have you gotten? Where are you stuck?

Please be complete. Thank you!