$(8$ points$)$ Let our universe be $\{4,6,8\}$ and predicate $P(x)$ denotes $lo{g}_2(x^3)8.$ Expand the

following statements and determine whether they are True or False. Justify your answer.

$($a$)$ AAxP$(x)$

$($b$)$ EExP$(x)$

$($c$)$ AAxnotP$(x)$

$($d$)$ notAAxP$(x)$

$(6$ points$)$ Let our universe be the set of all people, $P(x)$ denote $"x$ is a professor", and $C(x)$

denote $"x$ is a clown". Translate the following logical statements into English.

$($a$)$ AAx$(P(x)C(x))$

$($b$)$ AAx$(P(x)vvC(x))$

$($c$)$ EEx$(P(x{)}^{{?}^{?}}C(x))$