set notation for a set of ordered tuples

Question

For a set of tuples: $$D = \{(a_1', c_1',q_1), (a_2',c_2',q_2),...\}$$ where $a_i' \in A'$, $c_i' \in C'$, and $q \in Q$.
Note that, for example, $(a_1',c_2',q_1) \notin D$.

How can I write $D$ in terms of $A',C',Q$ ?

For example, the Cartesian product ($\times$) would give me the set of all combinations. That's not what I want. My question is, is there an expression for the element-wise product? For example, this paper writes something similar as $$D = \{<a_i, c_i, q>\}, i \in N$$ and a colleague suggests $$D = \{A',C',Q\}$$ which I think is incomplete.

Answer 1

What do you mean "how can I write $D$ in terms of $A'$,$C'$,$Q$"? From what I understand you have the three sets $A'$,$C'$,$Q$ with a labelling of their elements by $\mathbb{N}$, therefore $D=\{(a_i,c_i,q_i)\in A'\times C'\times Q : i \in \mathbb{N} \}$. This gives you explicitly the set $D$ whose first two elements you described. This is perfectly fine in mathematical notation, but without any more specific detail I do not think you can have a better description of $D$.

Answer 2

Note that you actually don't have three sets that should be combined but three sequences.

You could define a function $\mathrm{zip}_3$ taking three sequences $x, y, z$ and returning a new sequence: $$ \mathrm{zip}_3 : X^{\mathbb N} \times Y^{\mathbb N} \times Z^{\mathbb N} \to (X \times Y \times Z)^{\mathbb N}\\ \mathrm{zip}_3(x, y, z)(i) = (x(i), y(i), z(i)) $$ where $X, Y, Z$ are sets.

Functions like this exist in several programming languages like Haskell and Python, but are not very common in mathematics.