Understanding of definition in set theory I'm a computer scientist with just a basic understanding of set theory.
In the book "Theory of Modeling an Simulation", a multi component system (MC) is defined as a structure of the following:
$$M C =(T, X, \Omega, Y, D, \{M_d\})$$
With $T$ as a time base, $X$ as a set of input values, $\Omega$ as a set of input segments and $Y$ as a set of output values.
$M_d$ is defined as:
\begin{equation}
    M_d = (Q_d, E_d, I_d, \Delta_d, \Lambda_d)
\end{equation}
where $Q_d$ is the set of states of component $d$.
$I_d \subseteq D$ are all components influencing $d$.
$E_d \subseteq D$ are all components which are influenced by $d$.
$\Delta_d : \times_{i \in I_d}Q_i \times \Omega \rightarrow \times_{j\in E_d}Q_j$ is the state transition function of $d$.
$\Lambda_d : \times_{i_ \in I_d}Q_i \times \Omega \rightarrow Y$ is the output function of $d$.
Currently, I'm totally able to understand everything except the definition of  $\Delta_d$ and $\Lambda_d$. 
My issue is at the following: can you tell me what $\times_{i_ \in I_d}Q_i$ in the definitions from $\Delta_d$ and $\Lambda_d$ means? I just can not understand why there is a $\times$ sign in front is..
Even more, through my limited understanding, I do not know where to search for this.
Thank you for your help!
 A: The "$\times_{stuff}$" is an indexed Cartesian product - similar to "$\sum_{stuff}$" or "$\prod_{stuff}$." Basically, it's a fancy - and sometimes confusing - way of writing an unknown-length product all in one go. For example, if $A=\{1,2\}$, $B_1=\{3\}$, and $B_2=\{4,5\}$, then $$\times_{a\in A}B_a=\{(1\mapsto 3,2\mapsto 4), (1\mapsto 3,2\mapsto 5)\}.$$ This basically is $\{3\}\times\{4,5\}$, but "indexed" by $A$: it has two elements, one of which is the function sending $1$ to $3$ and sending $2$ to $4$ and the other of which is the function sending $1$ to $3$ and $2$ to $5$.


*

*Fully formally, "$\times_{a\in A}b_a$" refers to the set of all functions $f$ with domain $A$ such that $f(a)\in b_a$ for all $a\in A$. But this is probably needlessly confusing at first glance.



So for example, $\Delta_d$ takes as input $(1)$ an assignment of states to each component influencing $d$ and $(2)$ an input segment, and outputs an assignment of states to components influenced by $d$.
A: $\times_{i\in I_d}Q_i$ is the Cartesian product of all $Q_i$ where $i$ is in the set $I_d$. The Cartesian product is the set of tuples $(q_i)$ with $q_i\in Q_i$, for each $i\in I_d$. For example, if $I_d=\{1,2\}$, then $\times_{i\in I_d}Q_i=Q_1\times Q_2=\{(q_1,q_2):\ q_1\in Q_1\text{ and }q_2\in Q_2\}$.
The $\Delta_d$ is then a function that inputs tuples of elements of the $Q_i$, for $i\in I_d$, and the tuples has an extra last entry taken from $\Omega$, and outputs tuples of the $Q_j$, for $j\in E_d$.
Following the example above, if $E_d=\{3,4,5\}$, then $\Delta_d$ inputs tuples $(q_1,q_2,\omega)$, with $q_1\in Q_1$, $q_2\in Q_2$ and $\omega\in\Omega$, and outputs tuples $(q_3,q_4,q_5)$, with $q_3\in Q_3,q_4\in Q_4,q_5\in Q_5$.
A: $\times_{i_ \in I_d}Q_i$ is the cartesian product of all the sets $Q_i$.  Therefore, $\Lambda_d$ is a function whose inputs are a tuple of states and an input segment, and whose value is a tuple of states.
A: This is the notation for a Cartesian product. 
You might know this notation in another context, from your experience with geometry:


*

*$\mathbb R\times \mathbb R\times \mathbb R$, usually denoted in shorthand as $\mathbb R^3$ is the set of ordered triple $(x_1,x_2,x_3)$ such that $x_1 \in \mathbb R$ and $x_2 \in \mathbb R$ and $x_3 \in \mathbb R$. This is the mathematical notation for 3-dimensional Euclidean space.


For another example, 


*

*$\mathbb N \times \mathbb N \times \mathbb N$ similarly defined except that $x_1 \in \mathbb N$, etc. This is the notation for the set of "integer lattice points" as a subset of Euclidean 3-space.


In these examples, the Cartesian factors were all the same.
$\times_{i \in I} Q_i$ is the notation for a Cartesian product where the Cartesian factors are not required to be the same. For example:


*

*$Q_1 \times Q_2 \times Q_3$, also denoted $\times_{i \in I} Q_i$ where $I = \{1,2,3\}$, is the set of ordered triples $(x_1,x_2,x_3)$ such that $x_1 \in Q_1$ and $x_2 \in Q_2$ and $x_3 \in Q_3$.


It's particularly convenient to understand this when the index set $I$ is a set of consecutive integers starting with $1$ and ending with some intege $n$, i.e. $I = \{1,...,n\}$. But it's not necessary. You can actually use any index set $I$ whatsoever, and you can define Cartesian products with that index set. I think I won't go into this because it's somewhat technical, and because if you are only ever working with finite index sets then you can usually change your notation to work only with index sets of the form $\{1,...,n\}$.
