Implications of knowing how to order the elements of an $n$-tuple.

Question

Consider the ordered n-tuple $\{a_1,a_2,..., a_n\}$ with $a_i\in \mathbb{R}$ $\forall i=1,...,n$.

$\pi$-operator: Let $\pi$ be an operator that tells me

1) The position in the original n-tuple of $\{a_1,a_2,..., a_n\}$ when ordered from smallest to largest. When two elements of $\{a_1,a_2,..., a_n\}$ are equal we assume the convention that we put firstly the element coming first in $\{a_1,a_2,..., a_n\}$.

2) The relational operator (<,=) between the elements of $\{a_1,a_2,..., a_n\}$ when ordered from smallest to largest.

Examples:

• take $n=3$, $a_1=0$, $a_2=-100$, $a_3=4$; we have $a_2<a_1<a_3$; hence, $\pi(\{0,-2,4\})=\{\{2,1,3\}, \{<,<\}\}$

• take $n=3$, $a_1=0$, $a_2=100$, $a_3=0$; we have $a_1=a_3<a_2$; hence, $\pi(\{0,100,0\})=\{\{1,3,2\}, \{=,<\}\}$

Question: Suppose that I know $\pi(\{0,a,b\})$. Does this imply that I know $\pi(\{0,a,b,b-a\})$?

Answer 1

No, you don't nessecarily know $\pi(\{0,a,b,b-a\}) $.

Consider $a=1$ and $b=3$.$$ \pi(\{0,a,b\})=\{\{1,2,3\},\{<,<\}\} \\ \pi(\{0,a,b,b-a\})=\{\{1,2,4,3\},\{<,<,<\}\} $$

Consider $a=2$ and $b=3$.$$ \pi(\{0,a,b\})=\{\{1,2,3\},\{<,<\}\} \\ \pi(\{0,a,b,b-a\})=\{\{1,3,4,2\},\{<,<,<\}\} $$

These have the same values for $\pi(\{0,a,b\})$, but different values for $\pi(\{0,a,b,b-a\})$. Therefore $\pi(\{0,a,b,b-a\})$ cannot be determined from $\pi(\{0,a,b\})$.