Is this another way of saying T is a set of multiples of 3 in N?

Question

Is this another way of saying T is a set of multiples of 3 in N?

Answer 1

$T=3\Bbb N$ is a commonly used shorthand for your definition. But yes, your definition works. Just get rid of "$:t$ in there, and write "$T=\{\cdots\}$" directly.

Reading your question a bit more, if you just want to say that $T$ is some set containing only multiples of $3$, but not necessarily all of them, then you shouldn't use $\{\cdots\}$, since those are used for when you want to give an exact description of a specific set. Instead, say that all elements of $T$ are multiples of $3$:$$\forall t\in T\exists n\in \Bbb N(t=3n)$$