$($ The Question Proof all Mathematically.$)($rough set theory and the book an introduction to rugh set theory and its Application Author:zbigniew suraj$)$

Proof properties of the lower and upper approximation $?$

$R_{**}(x)subexsube{R}^{**}(x)$

$R_{**}(xY)=R_{**}(x)R_{**}(Y)$

$R_{**}(xY)supe{R}_{**}(x)R_{**}(Y)$

$R^{**}(xY)sube{R}^{**}(x)R^{**}(Y)$

xsubeY$R_{**}(x)sube{R}_s(Y)$&$R^{**}(x)sube{R}^{\text{'}}(Y)$

$R_{**}(-x)=-R^{**}(x)$

$R^{**}(-x)=-R_{**}(x)$

Proof the properties of the rough membership function $?$

${}_x^R(x)=1$ iff $xin{R}_{**}(x)$

${}_x^R(x)=0$ iff xinU$-R^{**}(x)$

$0<{}_x^R(x)<1$ iff $xinR{N}_R(x)$

${}_{U-x}^R(x)=1-{}_x^R(x)$ for any xinU

${}_{xY}^R(x)$max$({}_x^R(x),{}_Y^R(x))$ for any xinU

${}_{xY}^R(x)$min$({}_x^R(x),{}_Y^R(x))$ for any xinU