QUESTION $3[10]$Below a Tarski World is given followed by ten sentences. Which of the sentences are true and whichsentences are false in the given world?backleft rightfrontTarski World: Question $5$a: C$,$ Mf: D$,$ Sd: T$,$ Me: C$,$ S b: D$,$ L c: T$,$ L$4$Sentences:$1.$ x y $$SameSize$($y$,$ x$)2.$ x $($Tet$($x$)$ y z $($Cube$($y$)$ Tet$($z$)$ RightOf$($y$,$ x$)$ LeftOf$($x$,$ z$)))3.$ x y $($Adjoins $($x$,$ y$)$SameSize$($x$,$ y$))4.$ xy $[($Tet$($x$)$ Smaller$($x$,$ y$))$ Medium$($x$)]5.$ xy $[($Dodec$($x$)$ Dodec$($y$))($LeftOf$($x$,$ y$)$ RightOf$($x$,$ y$))]6.$ y x $($Medium$($y$)($Tet$($y$)($Cube$($x$)$ Smaller$($y$,$ x$))))7.$ xy$[($Tet$($x$)$ Cube$($y$))$ Larger$($x$,$ y$)]8.$ x y $($Cube$($x$)$ Tet$($y$)$SameSize$($x$,$ y$))9.$ x y $[$x y Tet$($x$)$ Tet$($y$)$ Medium$($x$)$ Medium$($y$)]10.$ Cube$($d$)$ x y SameShape$($x$,$ y$)$QUESTION $4[15]4\mathrm{.}1$ Transform the following formula into the prenex normal form: $(5)$x $(($C$($x$)$ y $($T$($y$)$ L$($x$,$ y$)))$ y $($D$($y$)$ B$($x$,$ y$)))4\mathrm{.}2$ Find the prenex normal form of x $($y R$($x$,$ y$)$ y $$S$($x$,$ y$)($y R$($x$,$ y$)$ P$))(5)4\mathrm{.}3$ Transform the following formula into prenex normal form: $(5)$x $($P$($x$)(($y$)($P$($y$)$ P$($f$($x$,$y$)))($y$)($Q$($x$,$y$)$ P$($y$))))$QUESTION $5[45]$In this question, you have to construct formal proofs using the natural deduction rules. The Fitch systemmakes use of these rules.A summary of the rules of natural deduction is given on pages $573$ to $578$ of your textbook. Consult thiswhen you do this question. Remember that De Morgans laws and other tautologies are not permissiblenatural deduction rules. You are also not allowed to use Taut Con, Ana Con or FO Con. It is important tonumber your statements, to indicate subproofs and at each step to give the rule that you are using.Hint: If you have access to a computer, take advantage of the fact and use Fitch$\mathrm{.}5\mathrm{.}1($Formally$)$ prove that the following two premises are contradictory: $(6)1.$ x Clever$($x$)2.$ x Clever$($x$)$COS$2661/105/0/20245\mathrm{.}2$ Using the natural deduction rules, give a formal proof of:xy$($Ixy Iyx$)$ xy$($Ixy Iyx $)(12)5\mathrm{.}3$ Construct a proof for the argument:xyz$[($Sxy Syz$)$ Sxz$],$ x$$Sxx xy$($Sxy $$Syx$)(13)5\mathrm{.}4$ Construct a proof for the argument:xyRxy xy$($RxyzRzx$),$ xy$($RyxzRxz$)$ xyRxy