| Assertion: A relation \[R=\left\{ \left( a,\,b \right)\,:\,\,|a-b|\,<\,2 \right\}\]defined on the set \[A=\left\{ 1,\,2,\,3,\,4,\,5 \right\}\] is reflexive. |
| Reason: A relation R on the set A is said to be reflexive if for \[\left( a,\,b \right)\in \,R\] and\[\left( b,\,c \right)\in \,R\], we have \[\left( a,\,c \right)\in R\]. |
A) Both A and R are individually true and R is the correct explanation of A.
B) Both A and R are individually true and R is not the correct explanation of A.
C) 'A' is true but 'R' is false
D) 'A' is false but 'R' is true
E) Both A and R are false.
Correct Answer: C
Solution :
Given \[R=\left\{ \left( a,\,b \right)\,\,:\,\,|\,a\,-\,b|\,<\,2 \right\}\] = {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (1, 2), (2, 1), (2, 3), (3, 2), (3, 4). (4, 3), (4, 5), (5, 4)}. Here \[\left( 1,\,1 \right)\in R\], \[\left( 2,\,\,2 \right)\in \,R\], \[\left( 3,\,\,3 \right)\,\in \,R\], \[\left( 4,\,\,4 \right)\in \,R\], \[\left( 5,\,5 \right)\in R\] \[\Rightarrow \]Relation R is reflexive on set \[A=\left\{ 1,\,2,\,3,\,4,\,5 \right\}\]\[\therefore \]Assertion A is true We know that relation R is reflexive if \[\left( a,\,a \right)\in R\,\,\forall \,a\in A\] \[\therefore \]Given Reason R is false Hence option [C] is the correct answer.
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