Equivalence Relations lists

Question

Let $X$ be a non-empty set and $n\in\mathbb N$. Then $X^n$ is the Cartesian product of $n$ copies of $X$. A relation can be defined on $X^n$ by $(a_1,a_2,\dots,a_n) \sim (b_1,b_2,\dots,b_n)$ if and only if every $x\in X$ appears the same the number of times in the first list as it does in the second.

Question:Let $X=\mathbb R^3$. For the relation above, list all elements which are related to $(0,1,2)$.

Would the answer to the question be all the possible ordered triplets, for example $(0,0,0), (0,0,1),\dots ,(1,1,1), \dots, (2,2,2), \dots, (1,0,2)$ etc.?

Answer 1

Your suggested answer is incorrect.

Your condition states that

every $x\in X$ must appear the same number of times in the first list as it does in the second.

So if $(a,b,c)\sim (0,1,2)$ then $(a,b,c)$ must contain one $0$, one $1$ and one $2$. This should be enough for you to figure out all the triples related to $(1,2,3)$.