Inverse Relation
Category : JEE Main & Advanced
Let A, B be two sets and let R be a relation from a set A to a set B. Then the inverse of R, denoted by \[{{R}^{-1}},\] is a relation from B to A and is defined by \[{{R}^{-1}}=\{(b,\,a):\,(a,\,b)\,\in R\}\]
Clearly \[(a,\,b)\in R\Leftrightarrow (b,\,a)\in {{R}^{-1}}\]. Also, Dom \[(R)=\,\,\,\text{Range}\,\,\,({{R}^{-1}})\] and Range \[(R)=\,\,\,\text{Dom}\,\,\,({{R}^{-1}})\]
Example : Let \[A=\{a,\,b,\,c\},\,B=\{1,\,2,\,3\}\] and \[R=\{(a,\,1),\,(a,\,3),\,(b,\,3),\,(c,\,3)\}\].
Then,
(i) \[{{R}^{-1}}=\{(1,\,a),\,\,(3,\,b),\,(3,\,c)\}\]
(ii) Dom \[(R)=\{a,\,\,b,\,\,c\}=\] Range \[({{R}^{-1}})\]
(iii) Range \[(R)=\{1,\,\,3\}=\] Dom \[({{R}^{-1}})\]
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