Question

Show that each of the relation $R$ in the set $A=\{x\in Z:0\le x\le 12\}$, given by is an equivalence relation. Find the set of all elements related to $1$ in each case.

$R=\left\{\right(a,b):a=b\}$

Answer 1

$A=\{x\in Z:0\le x\le 12\}=\{0,1,2,3,4,......,11,12\}$
$R=\left\{\right(a,b):a=b\}=\left\{\right(0,0),(1,1),(2,2),........,(11,11),(12,12\left)\right\}$
For any element $a\in A$, we have $(a,a)\in R$, since $a=a$.
$\therefore R$ is reflexive.
Now, let $(a,b)\in R$.
$\Rightarrow a=b$
$\Rightarrow b=a$
$\Rightarrow (b,a)\in R$
$\therefore R$ is symmetric.
Now, let $(a,b)\in R$ and $(b,c)\in R$.
$\Rightarrow a=b$ and $b=c$
$\Rightarrow a=c$
$\Rightarrow (a,c)\in R$
$\therefore R$ is transitive.
Hence, $R$ is an equivalence relation.
The elements in $R$ that are related to $1$ will be those elements from set $A$ which are equal to $1$.
Hence, the set of elements related to $1$ is $\left\{1\right\}$.