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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?

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    $\begingroup$ The empty relation, and the relation with one element. $\endgroup$ – Ethan Bolker May 18 '18 at 16:30
  • $\begingroup$ The empty set is also a subset of $A\times B$. You can call it the empty relation. $\endgroup$ – drhab May 18 '18 at 16:30
  • $\begingroup$ @Ispil When you added mathjax formatting you forgot to escape the "{" with "\{". $\endgroup$ – Ethan Bolker May 18 '18 at 17:10
  • $\begingroup$ Relation 1: A and B are related. i.e. $R= \{(1,2)\}$. Relation 2: A and B are not related. i.e. $R =\emptyset$. $\endgroup$ – fleablood May 18 '18 at 18:06
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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 $$ R_1 = \{\langle1,2\rangle\} \\ R_2 = \varnothing $$

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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.

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  • $\begingroup$ If they are not related then how can it be called a relation? $\endgroup$ – pranjal verma May 19 '18 at 14:04
  • $\begingroup$ Consider this relation: two people are related if the have ever had sex on the moon. That is a relation isn't it? Just because nobody (so far as we've been told) is actually related doesn't mean it's not a relation. $\endgroup$ – fleablood May 19 '18 at 15:02
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    $\begingroup$ In mathematics a relation between two sets is a subset of the their cartesian product. The empty set is a subset. So it is a relation. That is why there are $2^n$ relations; because there are $2^n$ subsets. If you want to define for yourself that relations can't be empty, that'd be consistant but then you'd have to claim there are $2^n-1$ and not $2^n$ relations for any cross product and you'd have to keep in mind every other mathematician is using a different definition than you are. $\endgroup$ – fleablood May 19 '18 at 15:06

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