Let's say $S = \{X_1, X_2, X_3, ...\}$ where each $X_i$ is some set.
For all $X,Y,Z \in S$, define an equivalence relation ~ by $((X,Y) \sim (X,Z)$ if $|Y| = |Z|)$. Or simply, the first element of the tuples is the same, and the second elements have the same size.
Let's say I want to now change how I define this equivalence relation ~ to
$((Y,X) \sim (Z,X)$ if $|Y| = |Z|)$. In other words, if the second element of the tuple is the same and the size of the first elements is equal.
Can I, somehow, parametrize my ~ so that it will work for both definitions?
Here is my naive approach as that of a non-mathematician:
Let $T = \{1,2\}$.
Define equivalence relation $R(t), t\in T$ by:
- $R(1) = ((X,Y) \sim (X,Z)$ if $|Y| = |Z|)$
- $R(2) = ((Y,X) \sim (Z,X)$ if $|Y| = |Z|)$
I want to then be able to say, that equivalence relation $R(1)$ splits $S$ into these equivalence classes, while $R(2)$ splits it into those. But at the same time I want to refer to an equivalence relation $R(t)$ for some $t \in T$ (since some of my results hold regardless of the type).
Your approach is fine. Equivalently, given any $t \in \{1, 2\}$, we can define the equivalence relation $R_t$ on $S^2$ by: $$ R_t = \{((A_1, A_2), (B_1, B_2)) \in S^2 \times S^2 \mid A_t = B_t \text{ and } |A_{3-t}| = |B_{3-t}|\} $$