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Relations and Functions — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSRelations and Functions3 Marks
Question
Determine whether the below relation is reflexive, symmetric and transitive:
Relation R in the set Z of all integers defined as
R = {(x, y) : x – y is an integer}
✓

Answer

It is given that Relation R in the set Z of all integers is defined as
R = {(x, y) : x – y is an integer}
Now, for every x $\in$ Z, (x, x) $\in$ R, as x - x = 0 is an integer.
$\Rightarrow$ R is reflexive.
Next, for every x, y $\in$ Z if (x, y) $\in$ R, then x - y is an integer.
$\Rightarrow$ -(x - y) is also an integer.
$\Rightarrow$ (y - x) is an integer.
$\Rightarrow$ (y - x) $\in$ R
$\Rightarrow$ R is symmetric.
Further, Take (x, y) ,(y, z) $\in$ R where x, y, z $\in$ R,
$\Rightarrow$ (x - y) and (y - z) are integers.
$\Rightarrow$ (x - z) = (x - y) + (y - x) is an integer.
$\Rightarrow$ (x, z) $\in$ R
$\Rightarrow$ R is transitive.
Therefore, R is reflexive, symmetric and transitive.

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