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I have some doubts regarding relations and binary relations in particular.This is what I understand :

1) The graph $G_R$ of a relation R on X and Y is the subset of X × Y defined by $G_R$ = {(x, y) ∈ X × Y : xRy} . This graph contains all of the information about which elements are related. Conversely, any subset G ⊂ X×Y defines a relation R by: xRy if and only if (x, y) ∈ G. Thus, a relation on X and Y may be (and often is) defined as subset of X × Y . Is it true that for sets, it doesn’t matter how a relation is defined, only what elements are related ?

2)Since any subset of A × B is a relation from A to B, it follows that the number of relations from A to B is $2^{|A×B|} = 2^{|A|·|B|}$.So, if A & B are null sets $\implies$ no. of relations is 1. So, will it lead to the statement that there exist at least one relationship between any two sets in the universe?

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  • $\begingroup$ Yes to all the questions. $\endgroup$ – Tobias Kildetoft Apr 8 '16 at 13:03
  • $\begingroup$ @TobiasKildetoft Are you implying that all the sets in the universe are related to each other ? I think this is only true for any undefined relation and not for any defined relation . For example :Suppose that S is a set of students enrolled in a university and B is a set of books in a library. We might define a relation R on S and B by: $s \in S $ has read $ b \in B$. In that case, sRb if and only if s has read b. But in case of an undefined relation we may relate any two sets by saying the null is subset of both. $\endgroup$ – katipra Apr 8 '16 at 13:17
  • $\begingroup$ I am saying that the answer to your question of whether there is a relation between any two sets is "yes". For example the empty relation. $\endgroup$ – Tobias Kildetoft Apr 8 '16 at 13:18
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1) yes

2) yes. If A and B are empty sets, then the empty set is a (and the only possible) relation between them

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