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?