Coming from engineering background, I'm getting a little lost in terminology and symbols, but I still want to be mathematically precise.
In engineering optimization, I often have say two design variables, where a variable is a set of possible alternative values that dictate the value of a design (a mapping from design space to objective space);
\begin{equation} \begin{aligned} X_1 &= {1, 3, 4} \ X_2 &= {1, red, black}\ \end{aligned} \end{equation}
This would define a design space $\Theta = \{X_1,X_2\}$. And $\Theta \rightarrow \Omega$, where $\Omega$ is the objective space. i.e. $\left|\Omega\right| = \Re$ in single objective optimization.
How do I express the "`size"' of this design space? I believe it's true to say that the cardinality (and magnitude? same term?) of the design space is $\left|\Theta\right| = 2$. But I am also concerned with the total number of design vectors in this finite space, say $\theta$. In this case, there are $\left|X_1\right| \times \left|X_2\right| = 9$ possible design vectors, or combinations of 2-tuples.
In general, for $m$ variables, the number of vectors is $\theta$;
\begin{equation} \begin{aligned} \theta = \prod_{i=1}^{m} \left|X_i\right| \end{aligned} \end{equation}
Is there a cleaner or more recognizable way to represent $\theta$? is the above description and choice of symbols common? I've looked at several texts on optimization, but haven't found any that really express it in this way.