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Relations and Functions — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSRelations and Functions2 Marks
Question
Determine whether the below relation is reflexive, symmetric and transitive:
Relation R in the set A of human beings in a town at a particular time given by
R = {(x, y) : x and y live in the same locality}
✓

Answer

Given that R = {(x, y) : x and y live in the same locality}
Clearly, (x, x) $\in$ R as x and x live in the same locality.
$\Rightarrow$ R is reflexive.
Now, if (x, y) $\in$ R, then x and y live in the same locality.
$\Rightarrow$ y and x live in the same locality.
$\Rightarrow$ (y, x) $\in$ R
$\Rightarrow$ R is symmetric.
Further, let (x, y), (y, z) $\in$ R
$\Rightarrow$ x and y live in the same locality and y and z live in the same locality.
$\Rightarrow$ x and z live in the same locality
$\Rightarrow$ (x, z) $\in$ R
$\Rightarrow$ R is transitive.
Therefore, R is reflexive, symmetric and transitive.

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