Finding all the relations of a given possible subsets.

Question

I am studying relations and I know that if the Cartesian product $A \times B$ has $n$ elements then the number of relations are $2^n$. Now let us take a set $A$ and $B$ such that $A =\{1\}$ and $B = \{2\}$. Then the Cartesian product $A \times B$ will be $\{(1,2)\}$ and it will have 1 element; but from the formula how can it have 2 relations?

Answer 1

There are two different relations between $A=\{1\}$ and $B=\{2\}$. We can call them $R_1$ and $R_2$, defined by $$ 1 \mathrel{R_1} 2 \quad\text{is }{\it true} \\ 1 \mathrel{R_2} 2 \quad\text{is }{\it false} $$ Remember that all we need to know about a relation $R$ between $\{1\}$ and $\{2\}$ is whether $1\mathrel R 2$ holds or not.

As sets, this would be $$ R_1 = \{\langle1,2\rangle\} \\ R_2 = \varnothing $$

Answer 2

There are $2^n$ relationships because given any pair $a,b$ either $a,b$ are related or they are not. So there are relations (half of them) in which they are related, and there are relations (half of them) in which that are not.

So if $1,2$ is the only pair then either they are or are not related. So there is one relation in which they are related. And there is another in which they are not related.

$R_1$ is $1$ and $2$ are related. So $(1,2) \in R_1$ and $ R_1 = \{(1,2)\}$.

$R_2$ is $1$ and $2$ are not related. And $(1,2)\not \in R_2$ and $R_2 = \emptyset$.

Semantically counter-intuitive, there is an "empty relation" where no elements are related to any other elements.