I would like to create a mathematical object which is a collection of vectors and matrices with different dimensions; for example
$P := \{\mathbf{a},\mathbf{B},\mathbf{c} \}$,
where $\mathbf{a} \in \mathbb{R}$, $\mathbf{B} \in \mathbb{R}^{6\times 6}$, and $\mathbf{c} \in \mathbb{R}^9$. Repetition is not allowed.
Question 1: is $P$ a set or a tuple?
Question 2: which notation do you recommend to differentiate sets and sequences? For example, I usually write sequences as
$\{a_1,a_2,a_3\}$,
which is the same notation of the sets. How do math researchers usually differentiate these two objects in their writting?
Thank you.
A set is a collection of objects in which the ordering of the objects does not matter, and it is denoted with curly braces such as $\{a,B,c\}$. Hence, $P$ is a set the way you have it defined above.
A tuple is a collection of objects in which the ordering of the objects does matter, and it is denoted with parentheses such as $(a,B,c)$.
NOTE: It is true that mathematicians sometimes use braces to denote a sequence, in which case the ordering does matter, and this seems to be the source of some confusion for you. The distinction outlined above is typically used in set theory. If you are working with both sets and sequences and want to distinguish the two properly, use parentheses around the sequence instead of braces, such as $(a_1,a_2,a_3,...)$. The reader, if a mathematician, will understand what you mean. If you really want to be clear, then preface your work with a statement like "Let $\{a\}$ be the set containing $a$, and let the $(a_n)$ be the sequence $a_n$." This will ensure that you and the reader are on the same page.