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SETS — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSSETS1 Mark
MCQ
Let $n$ be a fixed positive integer. Let a relation $R$ defined on $I\ ($the set of all integers$)$ as follows: $aRb$ if $ \frac{\text{n}}{(\text{a}-\text{b})}$, that is, if $a − b$ is divisible by $n,$ then, the relation $R$ is:
  • A
    Reflexive only
  • B
    Symmetric only
  • C
    Transitive only
  • ✓
    An equivalence relation
✓

Answer

Correct option: D.
An equivalence relation
$R$ is reflexive since for any integer $a$ we have $a - a = 0$ and $0$ is divisible by $n.$
Hence, $aR$ a $\forall$ a $\in I$.
$R$ is symmetric, let $aRb$.
Then by definition of $R, a - b = nk$ where $k \in I$.
Hence $b - a = (-k)n$ where $-k \in I$ and so $bRa$.
Thus we have shown that $aRb$
$\Rightarrow bRa$.
$R$ is transitive, let $aRb$ and $bRc$.
Then by definition of $R,$ we have $a - b = k_1​n$ and $b - c = k_2​n,$ where $k_1,k_2 \in I$.
It then follows that
$a - c = (a - b) + (b - c) = k_1n + k_2n = (k_1+​ k_2)n$
where $k_1+k_2​ \in I$

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