Class JEE Mathematics Sets, Relations, and Functions Q #1043
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4 Marks 2022 JEE Main 2022 (Online) 29th June Morning Shift MCQ SINGLE
Let a set $A = A_1 \cup A_2 \cup ..... \cup A_k$, where $A_i \cap A_j = \phi$ for $i \neq j$, $1 \le j, j \le k$. Define the relation R from A to A by $R = \{(x, y) : y \in A_i$ if and only if $x \in A_i, 1 \le i \le k\}$. Then, R is :
(A) reflexive, symmetric but not transitive.
(B) reflexive, transitive but not symmetric.
(C) reflexive but not symmetric and transitive.
(D) an equivalence relation.
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Correct Answer: D
Explanation
$R = \{(x, y) : y \in A_i, iff x \in A_i, 1 \le i \le k\}$

(1) Reflexive: $(a, a) \Rightarrow a \in A_i$ iff $a \in A_i$

(2) Symmetric: $(a, b) \Rightarrow a \in A_i$ iff $b \in A_i$. $(b, a) \in R$ as $b \in A_i$ iff $a \in A_i$

(3) Transitive: $(a, b) \in R$ & $(b, c) \in R$. $\Rightarrow a \in A_i$ iff $b \in A_i$ & $b \in A_i$ iff $c \in A_i$. $\Rightarrow a \in A_i$ iff $c \in A_i$. $\Rightarrow (a, c) \in R$.

Therefore, the relation is an equivalence relation.

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