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Given 5 numbers: $x_1, x_2, x_3, x_4, x_5 \in \mathbb N$

all the 4 operations: $+ - \times /$

a specific brackets pattern:

$\left(\left(\left(x_1 + x_2 \right) - x_3\right) \times x_4 \right) / x_5 = result$

I can permutate at will the five numbers and the 4 operations, I'm interested in the perumutations that give the same result, so with a equivalence relation such as:

Two permutations that return the same result are said equivalent under the basic mathematic laws: Commutative, Associative and Distributive.

Could be said that two solutions are equivalent if and only if they share the same set of numbers?

For example, the following equations are equivalent:

(((142 + 350) - 372) x 125) / 15 = 1000.0 (((142 + 350) - 372) / 15) x 125 = 1000.0 

What I mean is that the brackets are fixed but the numbers and the operator can be moved around at will.

Also, what happens if the condition changes to something like:

$ |\left(\left(\left(x_1 + x_2 \right) - x_3\right) \times x_4 \right) / x_5 - result| < \epsilon $

Is the conjecture still Ture/False?

Would this somewhat related to a $\delta$, where $\delta$ might be:

$\delta = min(\delta_{ij}) \quad \forall i,j, \: i \ne j$

with $\delta_{ij} = |x_i - x_j|$

Or nothing could be said?

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    @Tsuyoshi: Thanks! I feel a little dumb not having thought that myself. Anyway I've updated my question, and maybe you want to put your counter-example in an answer, so that I can accept it.2012-02-12

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(This is essentially my comments on the question, slightly rephrased.)

Could be said that two solutions are equivalent if and only if they share the same set of numbers?

No, this is not true. For example, consider two formulas (3−2)×6=6 and (6−3)×2=6. They are not equivalent in your sense, because if we forget the numbers and just consider them as (xyz and (zxy, they are different in general. They happen to be equal because of the particular choice of the numbers.

If you want a counterexample with five numbers and four operators, you can append other numbers and operators: for example, consider (((3−2)×6)+12)/9 and (((6−3)×2)+12)/9.