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On the set $Z$ of integers, relation $\mathrm{R}$ is defined as "a $\mathrm{R} \mathrm{b} \Leftrightarrow$ $\mathrm{a}+2 \mathrm{~b}$ is an integral multiple of $3^{\prime \prime}$. Which oneofthe following statements is correct for $\mathrm{R}$ ?
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The correct answer is:
$\mathrm{R}$ is an equivalence relation
The given relation is $\mathrm{aRb} \Leftrightarrow \mathrm{a}+2 \mathrm{~b}$, is an integral multiple of 3
In this relation $\mathrm{aRa} \Leftrightarrow \mathrm{a}+2 \mathrm{a}=3 \mathrm{a}$, an integral multiple of $3 . \mathrm{So}$, it is
reflexive $\mathrm{aRb} \Leftrightarrow \mathrm{a}+2 \mathrm{~b}$ and
$b R a=b+2 a+4 b-4 b$
$=2(a+2 b)-3 b$ is also an integral multiple of 3 . So, it issymmetric Let there be another value c, $\mathrm{bRc}=\mathrm{b}+2 \mathrm{c}$, be an integral multiple of 3 . Then $a \mathrm{Rc}=\mathrm{a}+2 \mathrm{c}$
So, $a R b+b R c=a+2 b+b+2 c=a+2 c+3 b$ is integral
multiple of 3 , hence, $\mathrm{a}+2 \mathrm{c}$ is also integral multiple $\mathrm{o}$ :
$=0, \mathrm{aRb}$ and $\mathrm{bRc} \Rightarrow \mathrm{aRc} . \mathrm{So}$, it is transitive
Therefore, relation is reflexive, symmetric as well transitive. Hence, $\mathrm{R}$ is an equivalence relation.
In this relation $\mathrm{aRa} \Leftrightarrow \mathrm{a}+2 \mathrm{a}=3 \mathrm{a}$, an integral multiple of $3 . \mathrm{So}$, it is
reflexive $\mathrm{aRb} \Leftrightarrow \mathrm{a}+2 \mathrm{~b}$ and
$b R a=b+2 a+4 b-4 b$
$=2(a+2 b)-3 b$ is also an integral multiple of 3 . So, it issymmetric Let there be another value c, $\mathrm{bRc}=\mathrm{b}+2 \mathrm{c}$, be an integral multiple of 3 . Then $a \mathrm{Rc}=\mathrm{a}+2 \mathrm{c}$
So, $a R b+b R c=a+2 b+b+2 c=a+2 c+3 b$ is integral
multiple of 3 , hence, $\mathrm{a}+2 \mathrm{c}$ is also integral multiple $\mathrm{o}$ :
$=0, \mathrm{aRb}$ and $\mathrm{bRc} \Rightarrow \mathrm{aRc} . \mathrm{So}$, it is transitive
Therefore, relation is reflexive, symmetric as well transitive. Hence, $\mathrm{R}$ is an equivalence relation.
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