| Assertion: A relation R = {(1, 1), (1, 3), (1, 5), (3, 1), (3, 3), (3, 5)} defined on the set A = {1, 3, 5) is transitive. |
| Reason: A relation R on the set A is said to be transitive if for \[\left( a,\text{ }b \right)\text{ }\in \,\,\text{R}\] and \[\left( a,\text{ c} \right)\text{ }\in \,\,\text{R}\], we have \[\left( b,\,\,c \right)\in \,\,R\]. |
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 = {(1, 1), (1, 3), (1, 5), (3, 1), (3, 3), (3, 5)} |
| Here \[\left( 1,\,\,1 \right)\in R,\,\left( 1,\,3 \right)\in R\Rightarrow \left( 1,\,\,3 \right)\in R\] |
| \[\left( 1,\,\,1 \right)\,\,\in \,\,\,R,\,\,\,\left( 1,\,\,\,5 \right)\,\,\in \,\,R\,\,\Rightarrow \left( 1,\,\,5 \right)\,\,\in \,\,R\] |
| \[\left( 1,\,\,3 \right)\,\,\,\in \,\,\,R,\,\,\left( 3,\,\,1 \right)\,\in R\Rightarrow \left( 1,\,\,1 \right)\,\,\in \,\,R\] |
| \[\left( 1,\,\,\,3 \right)\,\,\in \,\,R,\,\left( 3,\,\,3 \right)\,\,\in \,\,R\,\,\Rightarrow \,\,\left( 1,\,\,3 \right)\,\,\in \,\,\,R\] |
| \[\left( 1,\,\,3 \right)\,\,\in \,\,R,\,\left( 3,\,\,5 \right)\,\,\in \,\,R\Rightarrow \,\left( 1,\,\,5 \right)\,\,\in \,\,R\] |
| \[\Rightarrow \]Given relation R is transitive |
| \[\therefore \]Assertion [A] is true |
| Also given Reason is not true {By definition transition Relation} |
| \[\therefore \]Assertion [A] is true but Reason [R] is false Hence option [C] is the correct answer. |
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