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Is there a logical symbol for "is of the form" which can be used as a shorthand in a statement like:

"Any even natural number is of the form $2n$"

?

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    Not that I am aware of. "Of the form" is not a very precise statement in any case, since "form" may not be well-defined. We usually say that they are *equal* to something or other, or that they "can be written" or "expressed" as, and you say it, you don't abbreviate it.2011-07-20
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    $\forall n(\exists k(n=k+k)\rightarrow ...)$, and you can always add "Denote by $E(n)$ the formula $\exists k(n=k+k)$" at the start of your text.2011-07-20
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    I would convey something like this through an implication: $\forall k$ (if $k$ is an even natural number, then $\exists n(k=2n)$), or words to that effect.2011-07-20
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    Presumably you are going to have to state $n$ is a natural number and perhaps what *even* means2011-07-20
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    I should also note that the mathematical notations are far from having a uniform standard. Let alone a uniform language with consistent interpretation (for example $\wedge$ is used for both conjuction as well exterior product). So asking for a "logical symbol" is not a very good question (albeit expected to be asked at one point or another by most people in most fields). To add on that, my previous comment used addition while another could have used multiplication and a third could have used division. If you do not specify in what context it is almost impossible to answer such question.2011-07-20
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    Not as far as I know: "is of the form" is plenty short. One of its traditional uses in number theory has largely been replaced by the congruence notation. Anyway, we should concentrate more on being understood than on being brief.2011-07-20

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Something like "If $x$ is an even natural number then $\exists n \in \mathbb{N} : x=2n$"?