Question

Show that the relation $R(a,b)$ in set $A=\{x\in z:0\le x\le 2\}$ is Equivalence

$i\left)R\right(a,b):|a-b|$ is multiple of $4$

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

Answer 1

$A=\{0,1,2\}$ $A\times A=\left\{\begin{array}{ccc}(0,0)& (0,1)& (0,2)\\ (1,0)& (1,1)& (1,2)\\ (2,0)& (2,1)& (2,2)\end{array}\right\}$

(i) $R(a,b):(a-b)$ is multiple $\Rightarrow R=\left\{\right(0,0),(1,1),(2,2\left)\right\}$

$\therefore$ It is equivalence

(ii) $R(a,b):a=b\Rightarrow R=\left\{\right(0,0\left)\right(1,1\left)\right(2,2\left)\right\}$

$\therefore$ It is equivalence