Let $M$ be the Kripke structure $($:$W,$I,$J$:$)$ where $W,$I and $J$ are defined as follows:

$W=\{t,u,v,x,y,z\}$

I : PropVar $2^W$ given by:

$I(p)=\{x,y,z\},I(q)=\{x,y,t\},I(r)=\{y,t,u,z\}.$

$J$ : PName $2^{WW}$ given by:

$J(A)=\{(w,w)|winW\}\{(x,y),(x,z),(z,t),(y,v),(v,y),(v,x)\}$

$J(B)=\{(x,w)|winW\}\{(y,t),(z,t),(t,v)\}$

I. $($I$5$ point$)$ For each principal, draw a diagram $($with all features labeled$)$ to represent the

Kripke structure $M$ with respect to the principal.Let M be the Kripke structure hW$,$ I, Ji where W$,$ I and J are defined as follows:

$$ W $=\{$t$,$ u$,$ v$,$ x$,$ y$,$ z$\}$

$$ I : PropVar $->2$W given by:

I$($p$)=\{$x$,$ y$,$ z$\},$ I$($q$)=\{$x$,$ y$,$ t$\},$ I$($r$)=\{$y$,$ t$,$ u$,$ z$\}.$

$$ J : PName $->2$W$\backslash$times W given by:

J$($A$)=\{($w$,$ w$)|$w in W$\}\backslash$cup $\{($x$,$ y$),($x$,$ z$),($z$,$ t$),($y$,$ v$),($v$,$ y$),($v$,$ x$)\}$

J$($B$)=\{($x$,$ w$)|$w in W$\}\backslash$cup $\{($y$,$ t$),($z$,$ t$),($t$,$ v$)\}$

$1.(15$ point$)$ For each principal, draw a diagram $($with all features labeled$)$ to represent the

Kripke structure M with respect to the principal.