I feel like most definitions of “independence” are circular. Consider how we count the number of cards in a standard deck of cards: $|S \times R| = |S||R|$, where $S$ is the set of suits, and $R$ is the set of ranks. That is, $$S = \{\text{Hearts, Diamonds, Spades, Clubs}\}$$ and $$R = \{ 2,3,\dots,\text{King},\text{Ace} \}.$$ We know that $|S| = 4$ and $|R| = 13$, so $|S \times R| = |S||R| = 4 \cdot 13 = 52.$ In such an example, we define that they are independent, because they are disjoint subsets. But do we know that? In this example, it is “obvious.” What about examples where it is not obvious?
How do you define “independence” in combinatorics
combinationscombinatoricsdiscrete mathematicspermutationsprobability
Best Answer
Let's say you have a structure $S$. This structure is a combination of a few Attributes, each from a certain set of possible values.
If we let $A_1,...,A_n$ be the sets of the possible values for the corresponding attributes, then we can, pretty general, define the structure $S$ as follows:
$$ S:= \{\pmatrix{x_1\\\vdots\\x_n}\in ⨉_{i=1}^n A_i\mid P\pmatrix{x_1\\\vdots\\x_n} \} $$
Where $P$ is a predicate, i.e. it models our constraints, on which combinations of attributes are allowed.
Our goal, as usual in combinatorics, is determining $|S|$.
We say that the structure $S$ is independent of an attribute $A_i$ is (in the combinatoric sense), if: $$ \forall x_1\in A_1,...,x_n\in A_n,y_i\in A_i:\qquad P(x_1,...,x_i,...,x_n) = P(x_1,...,y_i,...,x_n) $$
Simply put this means that we don't need to look at the value of attribute $A_i$ to find out whether a specific instance of the structure is valid.
We therefore can fix a specific $x_i\in A_i$ (which exactly we choose doesn't matter), and define $$P': ⨉_{k=1\\i\neq i}^n A_k\to \{\text{True},\text{False}\}$$ via $$P'(x_1,...,x_{i-1},x_{i+1},...,x_{n}) = P(x_1,...,x_{i-1},x_i,x_{i+1},...,x_{n})$$
In terms of the cardinality, this then means the following: $$ |S| =|A_i|\cdot |\{\pmatrix{x_1\\...\\x_{i-1}\\x_{i+1}\\...\\x_{n}}\in ⨉_{k=1\\k\neq i}^n A_k\mid P'\pmatrix{x_1\\...\\x_{i-1}\\x_{i+1}\\...\\x_{n}} \}| $$
As for every tuple $\pmatrix{x_1\\...\\x_{i-1}\\x_{i+1}\\...\\x_{n}}$, the structure instance $\pmatrix{x_1\\...\\x_{i-1}\\x_i\\x_{i+1}\\...\\x_{n}}$ is either valid for all choises of $x_i$, or for none of it.
To finish this, let's look at your example.
Our structure is the set of valid cards, where each card has the two attributes suit and rank.
Therefore we have $n=2$, $A_1:= \{\text{Hearts, Diamonds, Spades, Clubs}\}, A_2:=\{ 2,3,\dots,\text{King},\text{Ace} \}$.
Since we have no restrictions on our set members of $S$ besides that we have to pick from every attribute, we have further $P(x_1,x_2)=\text{True}$.
So, our structure $S$ is in this case independent of both $A_1$ and $A_2$, and therefore we have $|S| = |A_1|\cdot |A_2|$