We define a relation S on the set of all integers by: $nSk$ iff $n^2$ $=$ $k^2$ Decide if S is an equivalence relation. If so, what is the equivalence class of $9$?
It can be proven that S is an equivalence relation. What is the equivalence class of 9? I though it was {-81,81}, According to the key, it is {-9,9}. Am I wrong? If so, why?
Yes, sorry to say, you were wrong. First of all, no matter what the equivalence relation, the equivalence class of anything contains that thing. So $[9]_S$, the $S$-equivalence class of $9$, can't possibly be $\{-81, 81\}$. The $S$-equivalence class of $9$ is all things $x$ such that $9 S x$, which is to say, $[9]_S = \{x\in \mathbb{Z} \mid x^2 = 9^2\} = \{x\in \mathbb{Z} \mid x^2 = 81\} = \{-9, 9\}$.