JEE MAIN - Mathematics (2022 - 29th June Morning Shift - No. 9)
Let a set A = A1 $$\cup$$ A2 $$\cup$$ ..... $$\cup$$ Ak, where Ai $$\cap$$ Aj = $$\phi$$ for i $$\ne$$ j, 1 $$\le$$ j, j $$\le$$ k. Define the relation R from A to A by R = {(x, y) : y $$\in$$ Ai if and only if x $$\in$$ Ai, 1 $$\le$$ i $$\le$$ k}. Then, R is :
reflexive, symmetric but not transitive.
reflexive, transitive but not symmetric.
reflexive but not symmetric and transitive.
an equivalence relation.
Explanation
$$R = \{ (x,y):y \in {A_i},\,iff\,x \in {A_i}\,1 \le i \ge k\} $$
(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.
$$\Rightarrow$$ RElation is equivalnece.
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