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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.?

dbmag9
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Emma math
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1 Answers1

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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 $(1,2,3)$.

dbmag9
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  • @EmmaMath If you found this answer useful, you can upvote it using the arrow to the left, and accept it using the tick mark to indicate that it answered your question. – dbmag9 Nov 25 '20 at 09:12