# Shorthand for “has property…”

Is the mathematical shorthand for the phrase "has the property..." or "satisfies the condition..."? As in, when you have a set of properties or conditions predefined, how can you write that an object has those properties or satisfies those conditions?

For example, in Rudin's Principles of Mathematical Analysis, in the Appendix to "The Real and Complex Number Systems," Rudin defines three properties, (I), (II), and (III) that define a cut. Later on, he tries to prove that various sets satisfy these conditions and are therefore cuts. In Step 3, for instance, he defines a set $\gamma$, then proves that this set satisfies (I), (II), and (III) and is therefore a cut. Is there any symbolic notation for "$\gamma$ satisfies (I)" ?

Thanks

• I usually go with fulfills but I am not a native English speaker, so you should not trust me too much. – Jack D'Aurizio Jul 6 '17 at 14:11
• If the "property" $S$ is defined for some sort of "objects" and $\gamma$ is such kind of object, then $S(\gamma)$ will do. If $\text {Even}$ is defined for natural numbers, we can write: $\text {Even} (2)$. – Mauro ALLEGRANZA Jul 6 '17 at 14:22
• The short answer is no, especially for a situation with roman numerals like you quoted. But in mathematical logic there is a related, if more formal notion, of a model satisfying a statement (or set of statements) in some language; there we would write $\mathcal{M}\vDash \Gamma$. But no one would use this turnstile notation in the course of ordinary mathematical writing like the sort you quote from Rudin. – symplectomorphic Jul 6 '17 at 14:23

## 1 Answer

I've never stumbled across some standard, agreed-upon terminology. If you want to make it formal, you make the property a set $S$, and to say "$\gamma$ satisfies [the property]", you say $\gamma \in S$.

If you want formal, universally understood terminology, making the property a set is the way to go. If you want easily human-readable text, then substituting symbols, shorthand, or jargon in for phrases is not advisable. If you want succinct, often defining a set does not require much more space than defining the property itself.