$(12$pts$)$ Using logical equivalences, determine if the following are tautologies, contradictions or contingencies.

State the rules of logic that you use in each step. Do not apply more than two laws per step. Note that you

cannot use truth tables to justify your answers.

$($a$)(a^{{?}^{?}}$notb$)harr(ab)$

$($b$)$ not$(pvvq)vv(not{p}^{{?}^{?}}q)vvnotp$

$($c$)((pq{)}^{{?}^{?}}p)(q^{{?}^{?}}q)$

$($d$)(a$notb${)}^{{?}^{?}}(not(a^{{?}^{?}}b)$notc$)$

$(4$pts$)$ Let $p$ and $q$ be two propositions. Let us define new binary operators: uarr and darr.

Let puarrq$-=$not$(p^{{?}^{?}}q)$ and pdarrq$-=$not$(pvvq).$ Is it true that puarrq$-=$notpdarrnotq $?$

$(8$pts$)$ Translate the following into English, where $R(x)$ is $"x$ is a rabbit", $H(x)$ is $"x$ hops" and $M(x)$ is $"x$

knows math". The universe of discourse is all animals.

$($a$)$ EExnotH$(x)$

$($b$)$ AAx$(R(x{)}^{{?}^{?}}M(x))$

$($c$)$ AAx$((H(x{)}^{{?}^{?}}$notM$(x))R(x))$

$($d$)$ notEEx$(R(x{)}^{{?}^{?}}(H(x)vvM(x)))$AAxnotR$(x)$

$(12$pts$)$ Convert the following English statements into predicate logic expressions. Be sure to state your predi$-$

cates, for example: "Let $D(x)$ be the predicate $"$x can dance"". The universe of discourse is all humans.

$($a$)$ Not everyone loves Math.

$($b$)$ Every student met the assignment deadline.

$($c$)$ At least one person who does not bike owns a bicycle.

$($d$)$ Everyone but Mariyka came to class yesterday.

$($e$)$ Not all who dream are asleep.

$($f$)$ There is exactly one girl who can do miracles.