JEE Main & Advanced Mathematics Sets Definition

Let A and B be two non-empty sets, then every subset of \[A\times B\] defines a relation from A to B and every relation from A to B is a subset of \[A\times B\].

Let \[R\subseteq A\times B\] and \[(a,\,\,b)\in R\]. Then we say that \[a\] is related to \[b\] by the relation \[R\] and write it as \[a\,R\,b\]. If \[(a,\,b)\in R\],  we write it as \[a\,R\,b\].

(1) Total number of relations : Let A and B be two non-empty finite sets consisting of m and n elements respectively. Then \[A\times B\] consists of mn ordered pairs. So, total number of subset of \[A\times B\] is \[{{2}^{mn}}\]. Since each subset of \[A\times B\] defines relation from A to B, so total number of relations from A to B is \[{{2}^{mn}}\]. Among these \[{{2}^{mn}}\] relations the void relation \[\phi \] and the universal relation \[A\times B\] are trivial relations from A to B.

(2) Domain and range of a relation : Let R be a relation from a set A to a set B. Then the set of all first components or coordinates of the ordered pairs belonging to R is called the domain of R, while the set of all second components or coordinates of the ordered pairs in R is called the range of R.

Thus, Dom \[(R)=\{a\,:(a,\,b)\,\in R\}\] and Range \[(R)=\{b\,:(a,\,\,b)\,\in R\}\].