Let $A$ and $B$ be two finite sets with cardinalities $m$ and $n$, respectively. The total number of possible relations $R \subseteq A \times B$ between $A$ and $B$ is $2^{m \times n}$, since each relation is a subset of the Cartesian product $A \times B$.
However, I am interested in understanding these relations in terms of their structure rather than just counting all possible relations.
Example 1
Let $A = \{a_1, a_2\}$ and $B = \{b\}$. Hence, there are the following $2^{2 \times 1} = 4$ possible relations:
- $R_1 = \emptyset$.
- $R_2 = \{(a_1, b)\}$.
- $R_3 = \{(a_2, b)\}$.
- $R_4 = \{(a_1, b), (a_2, b)\}$.
However, in terms of structure, there are only 3 kinds of relations $S \subseteq A \times B$:
- $S_1$ which has the structure of $R_1$.
- $S_2$ which has the structure of $R_2$ and $R_3$.
- $S_3$ which has the structure of $R_4$.
This is because $R_2 = \{(a_1, b)\}$ and $R_3 = \{(a_2, b)\}$ are structurally the same, differing only in which element of $A$ is related to $b$.
Example 2
Now, let $B = {b_1, b_2}$ while keeping $A = {a_1, a_2}$. The total number of relations is $2^{2 \times 2} = 16$. The possible relations are:
$R_1 = \emptyset$.
$R_2 = \{(a_1, b_1)\}$.
$R_3 = \{(a_1, b_2)\}$.
$R_4 = \{(a_2, b_1)\}$.
$R_5 = \{(a_2, b_2)\}$.
$R_6 = \{(a_1, b_1), (a_1, b_2)\}$.
$R_7 = \{(a_2, b_1), (a_2, b_2)\}$.
$R_8 = \{(a_1, b_1), (a_2, b_1)\}$.
$R_9 = \{(a_1, b_2), (a_2, b_2)\}$.
$R_{10} = \{(a_1, b_2), (a_2, b_1)\}$.
$R_{11} = \{(a_1, b_1), (a_2, b_2)\}$.
$R_{12} = \{(a_1, b_1), (a_1, b_2), (a_2, b_1)\}$.
$R_{13} = \{(a_1, b_1), (a_1, b_2), (a_2, b_2)\}$.
$R_{15} = \{(a_1, b_1), (a_2, b_1), (a_2, b_2)\}$.
$R_{14} = \{(a_1, b_2), (a_2, b_1), (a_2, b_2)\}$.
$R_{16} = \{(a_1, b_1), (a_1, b_2), (a_2, b_1), (a_2, b_2)\}$.
In terms of structure, however, the distinct kinds of relations $S \subseteq A \times B$ are:
- $S_1$ which has the structure of $R_1$
- $S_2$ which has the structure of $R_2$-$R_5$.
- $S_3$ which has the structure of $R_6$-$R_7$.
- $S_4$ which has the structure of $R_8$-$R_9$.
- $S_5$ which has the structure of $R_{10}$-$R_{11}$.
- $S_6$ which has the structure of $R_{12}$-$R_{15}$
- $S_7$ which has the structure of $R_{16}$.
More generally
Here is the data I have gathered (by drawing case by case) on the number of relations of similar structure.
- For $m=n=1$: 2 relations grouped in 2 kinds.
- For $m=1$ and $n=2$: 4 relations grouped in 3 kinds.
- For $m=1$ and $n=3$: 8 relations grouped in 4 kinds.
- For $m=n=2$: 16 relations grouped in 7 kinds.
- For $m=2$ and $n=3$: 64 relations grouped in 13 kinds.
- For $m=n=3$: still checking.
Questions:
- What is the general method for determining the number of such distinct kinds of relations $S \subseteq A \times B$ based on structure for arbitrary $m$ and $n$?
- How does this classification extend to cases with more than two sets, such as relations $S \subseteq A_1 \times ... \times A_n$?
- Is there any algorithm that can allow me to determine exactly the form of those relations?
- Is there a specific mathematical term for this kind of classification, and in which field of mathematics is this topic studied in detail? (Any recommended textbooks or references would be greatly appreciated!)
