# Set terminology and symbols in optimization

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{aligned} X_1 &= {1, 3, 4} \ X_2 &= {1, red, black}\ \end{aligned}

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{aligned} \theta = \prod_{i=1}^{m} \left|X_i\right| \end{aligned}

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.

-

If you want to present the "size" with terminology rather than in symbolic form, you can refer to the design space as the Cartesian product of the design variables. In your example there are two design variables, each with three value alternatives, and in the absence of any constraints that prevent arbitrarily pairing those values, the Cartesian product gives the design space with a total of $3 \times 3 = 9$ possibilities.