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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. –  Arturo Magidin Jul 20 '11 at 20:40
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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. –  Asaf Karagila Jul 20 '11 at 20:42
    
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. –  Andres Caicedo Jul 20 '11 at 20:46
    
Presumably you are going to have to state $n$ is a natural number and perhaps what even means –  Henry Jul 20 '11 at 20:47
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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 One of its traditional uses in number theory has largely been replaced by the congruence notation.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. –  André Nicolas Jul 20 '11 at 22:39
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closed as too localized by Asaf Karagila, Jonas Teuwen, Eric Naslund, t.b., Zev Chonoles Dec 7 '11 at 4:54

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1 Answer

up vote 2 down vote accepted

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

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