Abbreviate a tuple with a variable?

Question

In a book I found the following notation $M = \{(x_1,y_1,z_1), \ldots, (x_n,y_n,z_n)\}$ for a set of 3-tuples. The author always refers to a tuple by writing $(x_i,y_i,z_i) \in M$.

I'm wondering if I can abbreviate the tuple with $T = (x_1,y_1,z_1)$ and then write the set as $M = \{T_i, \ldots, T_n\}$ and $T_i \in M$ to refer to an element? I never see someone abbreviating a tuple with a variable, so I'm asking here.

If the above is ok, would you use an uppercase letter like $T$ for the abbreviation of the tuple or a lowercase letter like $t$ (so e.g. $M = \{t_i, \ldots, t_n\}$)?

Answer 1

I'm quite sure you have, whether you noticed it or not. For instance, a vector is a good example of a tuple being treated as a variable. In general, in mathematics, it is quite common to use notations to refer to pretty much any mathematical object. As for which notation to choose, vectors are most commonly written with bold letters, as in $\mathbf t$, and sometimes underlined, as in $\underline t$. Especially in advanced literature, it is however common to simply use the unbold $t$.

Answer 2

There is no such hard and fast rule. But usually we tend to use capital letters for denoting sets, linear transformations etc. Thats just for the sake of avoiding confusion. I prefer $t=(x_1,y_1,z_1)$ to $T=(x_1,y_1,z_1)$.