Show that each of the relation $R$ in the set $A =\{x \in Z : 0 \leq x \leq 12\},$ given by $R =\{(a, b):|a-b| $ is a multiple of $4\}$
Show that each of the relation $R$ in the set $A =\{x \in Z : 0 \leq x \leq 12\},$ given by $R =\{(a, b):|a-b| $ is a multiple of $4\}$
Set $A=\{x \in Z: 0 \leq x \leq 12\}=\{0,1,2,3,4,5,6,7,8,9,10,11,12\}$
$R =\{( a , b ):| a - b | $ is a multiple of $4\}$
For any element, $a \in A$, we have $(a, a) \in R$ as $|a-a|=0$ is a multiple of $4.$
$\therefore R$ is reflexive.
Now, let $(a, b) \in R \Rightarrow|a-b|$ is a multiple of $4$
$\Rightarrow|-(a-b)|=|b-a|$ is a multiple of $4$
$\Rightarrow(b, a) \in R$
$\therefore R$ is symmetric.
Now, let $(a, b),\,(b, c) \in R$
$\Rightarrow|a-b|$ is a multiple of $4$ and $|b-c|$ is a multiple of $4$
$\Rightarrow(a-b)$ is a multiple of $4$ and $(b-c)$ is a multiple of $4$
$\Rightarrow(a-c)=(a-b)+(b-c)$ is a multiple of $4$
$\Rightarrow|a-c|$ is a multiple of $4$
$\Rightarrow(a, c) \in R$
$\therefore R$ is transitive.
Hence, $R$ is an equivalence relation.
The set of elements related to $1$ is $\{1,5,9\}$ as
$|1-1|=0$ is a multiple of $4$
$|5-1|=4$ is a multiple of $4$
$|9-1|=8$ is a multiple of $4$
Similar Questions
Let $R$ be a relation on the set of all natural numbers given by $\alpha b \Leftrightarrow \alpha$ divides $b^2$.
Which of the following properties does $R$ satisfy?
$I.$ Reflexivity $II.$ Symmetry $III.$ Transitivity
Let $R$ be a relation on the set of all natural numbers given by $\alpha b \Leftrightarrow \alpha$ divides $b^2$.
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