Problem $9.$ Existential, Universal Qualifiers & Predicates $(12$ points$)$

Use the following predicates:

circle$(x)$ is True when $x$ is a circle

rect$(x)$ is True when $x$ is a rectangle

star$(x)$ is True when $x$ is a star

shade$(x)$ is True when $x$ is shaded in

next$\_$to $(x,y)$ is True when the squares containing $x$ and $y$ touch $($i$.$e$.$ they're immediate neighbors left$-$right, top$-$bottom, or they share a corner diagonally$).$

to determine if each of the statements below is true or false. Provide a brief justification towards your answer. For example, given the statement rect$(b)$ a correct response is$,$ 'This is false as $b$ is not a rectangle'.

$($a$)[2$ pts$.]$ EExrect$(x)$

Solution:

$($b$)[2$ pts$.]$ EExnotshade$(x{)}^{{?}^{?}}$circle$(x)$

Solution:

$($c$)[2$ pts$.]$ star $(x)$notshade$(x)$

Solution:

$($d$)[2$ pts$.]$ shade $(x)$notstar$(x)$

Solution:

$($e$)[2$ pts$.]$ EExAAy$(notrect(y{)}^{{?}^{?}}$shade$(y))nex{t}_{t}o(x,y)$

Solution: