JEE MAIN - Mathematics (2023 - 8th April Evening Shift - No. 5)
Let $$\mathrm{A}=\{1,2,3,4,5,6,7\}$$. Then the relation $$\mathrm{R}=\{(x, y) \in \mathrm{A} \times \mathrm{A}: x+y=7\}$$ is :
reflexive but neither symmetric nor transitive
transitive but neither symmetric nor reflexive
symmetric but neither reflexive nor transitive
an equivalence relation
Explanation
Here, $A=\{1,2,3,4,5,6,7\}$
Since, $x+y=7 \Rightarrow y=7-x$
So, $\mathrm{R}=\{(1,6),(2,5),(3,4),(4,3),(5,2),(6,1)\}$
$\because(a, b) \in \mathrm{R} \Rightarrow(b, a) \in \mathrm{R}$
$\therefore \mathrm{R}$ is symmetric only.
Since, $x+y=7 \Rightarrow y=7-x$
So, $\mathrm{R}=\{(1,6),(2,5),(3,4),(4,3),(5,2),(6,1)\}$
$\because(a, b) \in \mathrm{R} \Rightarrow(b, a) \in \mathrm{R}$
$\therefore \mathrm{R}$ is symmetric only.
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