Class JEE Mathematics Sets, Relations, and Functions Q #1076
KNOWLEDGE BASED
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4 Marks 2005 AIEEE MCQ SINGLE
Let $R = \{(3, 3), (6, 6), (9, 9), (12, 12), (6, 12), (3, 9), (3, 12), (3, 6)\}$ be a relation on the set $A = \{3, 6, 9, 12\}$. The relation is :
(A) reflexive and symmetric only
(B) an equivalence relation
(C) reflexive only
(D) reflexive and transitive only
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Correct Answer: D
Explanation
To determine the properties of the relation $R$, we need to check for reflexivity, symmetry, and transitivity.

Reflexivity: A relation is reflexive if $(a, a) \in R$ for all $a \in A$. Since $(3, 3), (6, 6), (9, 9),$ and $(12, 12)$ are all in $R$, the relation is reflexive.

Symmetry: A relation is symmetric if whenever $(a, b) \in R$, then $(b, a) \in R$. We have $(6, 12) \in R$, but $(12, 6) \notin R$. Thus, the relation is not symmetric.

Transitivity: A relation is transitive if whenever $(a, b) \in R$ and $(b, c) \in R$, then $(a, c) \in R$. We have $(3, 6) \in R$ and $(6, 12) \in R$, and $(3, 12) \in R$. Also, $(3, 6) \in R$ and $(6, 6) \in R$, and $(3, 6) \in R$. The relation appears to be transitive.

Therefore, the relation $R$ is reflexive and transitive, but not symmetric.

So, the correct answer is Option D: $R$ is reflexive and transitive only.

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