Problem $3[24$ pts $(3$ each$)]$: Tarski World Predicates & Quantifiers

Using the following predicates:

square$($x$)$ is true if x is a square $($otherwise it is false$)$

star$($x$)$ is true if x is a star $($otherwise it is false$)$

circ$($x$)$ is true if x is a circle $($otherwise it is false$)$

shade$($x$)$ is true if x is shaded $($otherwise it is false$)$

next to$($x$,$ y$)$ is true if x and y are adjacent horizontally, vertically or diagonally. No object is next to itself.

For each of the statements below:

$$ write a simple, equivilent statement in english as the one given $$ determine whether the statement is true or false

i star$($c$)$shade$($c$)$

ii $$xcirc$($x$)$shade$($x$)$

iii $$xsquare$($x$)->$shade$($x$)$

iv $$x$$y$($star$($x$)$shade$($x$)$ next to$($x$,$ y$))->($shade$($y$)$ circ$($y$))$

v $$x$$ynext to$($x$,$y$)$

vi $$y$$xshade$($x$)$next to$($x$,$y$)$