4
$\begingroup$

Is the number of relations on $A \times B$ the same as the number of relations from $A$ to $B$?

Can anybody clear this doubt with some examples? In my notes, I have written the number of relations on $A \times B$ as $2^{(mn)^2}$ and number of relations from $A$ to $B$ as $2^{mn}$. But some friends are arguing both are same and answer is $2^{mn}$.

$\endgroup$
1
  • $\begingroup$ Welcome to MathSE. This tutorial explains how to typeset mathematics on this site. $\endgroup$ Oct 20, 2021 at 10:48

1 Answer 1

5
$\begingroup$

I think you are right.


A relation from $A$ to $B$ is a subset of $A\times B$.

A (binary) relation on a set $X$ is a relation from $X$ to $X$ hence is a subset of $X\times X$.

So a (binary) relation on set $A\times B$ is a subset of $(A\times B)\times(A\times B)$.


See here for example.

I can imagine though that sometimes mathematicians are kind of sloppy by the use of this terminology. So do not put too much trust in it and inform.

$\endgroup$
3
  • $\begingroup$ Any objections to this Statement if no I will consider this as ok $\endgroup$
    – Aayat khan
    Oct 21, 2021 at 14:01
  • $\begingroup$ Aayat. Of course you can wait for a while, but chances are small (I think) that someone will react on your comment by raising an objection. $\endgroup$
    – drhab
    Oct 21, 2021 at 14:48
  • 1
    $\begingroup$ Done ✅ thanks for the help $\endgroup$
    – Aayat khan
    Oct 22, 2021 at 6:50

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .