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I find that I am using the term "by the definition of" a lot in my proofs, especially after && in align* environments. Is there a symbol that I can use instead for conciseness?

Example:

Having established that $x$ is odd, I would say $x=2k+1$ for some $k\in\mathbb{Z}$, $\textit{by the definition of odd}$.

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    $\begingroup$ Can you give an example? $\endgroup$
    – Darius
    Jun 5, 2019 at 2:10
  • $\begingroup$ @MureyTasroc: Please include your example in the question itself. Comments are easily overlooked. $\endgroup$
    – Blue
    Jun 5, 2019 at 2:20
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    $\begingroup$ "it is trivial to see -", "even the dullest of readers can appreciate that -" are often good substitutes $\endgroup$
    – kyary
    Jun 5, 2019 at 2:29
  • $\begingroup$ "by the definition of" is logically equivalent to "if and only if" $\endgroup$
    – logo
    Jun 5, 2019 at 2:34
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    $\begingroup$ $:=$ (is equal to by definition). Is that what you're looking for? $\endgroup$
    – NoName
    Jun 5, 2019 at 3:02

1 Answer 1

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Usually, if something follows by definition in a (well-written) mathematical text you don't need to call that out. Your example (and I appreciate it's probably contrived) is actually a good example of that: having established that $x$ is odd you can just write that $x=2k+1$ without saying "by definition of odd" (by the way, either oddness, or oddity if you want to be mildly amusing, but "odd" alone is wrong).

If you want to remind the reader that something was defined further up, before something interesting/complicated was discussed, then slipping "(by definition)" in is sufficient to indicate that if they don't remember it they should go back and check.

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