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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

fangye
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  • My guess is that, strictly speaking, they're not necessary? Anyone familiar with the notion of relations would be able to get from "For $x,y \in X$, define $x \sim y \iff x -y \in M$" the notion that $\sim$ is a relation on $X$. But at the same time, I don't see why they would merit removal. Sure, it might be "obvious" or "redundant," but clarity never hurts either... – PrincessEev Nov 27 '19 at 02:26
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    It might be better to ask your professor - it's a matter of style, to a large degree. (At the least, the professor is saying that just writing "define" followed by the equation is sufficient, though it's also reasonable to argue that "a relation $\sim$ on $X$ by" could be useful if that clarification is useful) – Milo Brandt Nov 27 '19 at 02:26

1 Answers1

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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 $x \sim y \iff x - y \in M$.

Magdiragdag
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  • You may be right; I will ask my professor if they meant it. By the way, the first one is what I wrote in the beginning, which my professor did not accept. – fangye Nov 29 '19 at 20:47
  • @fangye Note the location of the universal quantification ('For (all) $x,y$') in 'For $x, y$, define $\sim$ by $P(x,y)$' v.s. 'Define $\sim$ by $P(x,y)$ for all $x, y$'. – Magdiragdag Dec 01 '19 at 10:40