May I write something like "Define a relation $\sim$ on a set by $x\sim y\iff x-y\in M$"?

Question

When stating the definition of quotient spaces, I wrote

Let $X$ be a linear space and $M$ be a linear subspace of $X$. For $x,y\in X$, define a relation $\sim$ on $X$ by \begin{align*} x\sim y\iff x-y\in M. \end{align*}

My professor circled the words "a relation $\sim$ on $X$ by" and asked me to remove them. But I did not know why these words must be removed.e

Answer 1

Your formulation may suggest that for every pair $x,y$ you define a relation $\sim$; instead, what you want to say is that you define a relation $\sim$ by stating how it behaves for every pair $x,y$. Maybe your professor would (also) have accepted

Define a relation $\sim$ on $X$ by $x \sim y \iff x - y \in M$, for every $x, y \in X$.

or something similar to your your own formulation in the title

Define a relation $\sim$ on $X$ by $x \sim y \iff x - y \in M$.