Equivalence relation notation

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Suppose that we want to define an equivalence relation $\sim$ on some space $X$ by $x_1 \sim x_2$ if $x_1 = -x_2$. Is it correct to just define $\sim$ by just saying $x_1 \sim -x_2$?

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No. If the relation $\sim$ is defined by $x_1 \sim x_2$ when $x_1 = -x_2$, then all you can say is $x_1\sim x_2$.

Because to say $x_1 \sim -x_2$ means $x_1 = -(-x_2)$, which is $x_1 = x_2$