$(6$pts$)$ Determine which of the following statements are True and explain why or why not. See the sample

solution in part $($a$).$

$($a$)12>10$ or unicorns exist.

Solution: True.

Let $p$ be the proposition $"12>10"$

Let $q$ be the proposition "unicorns exist"

The statement is thus translated into $pvvq.$

We know that $p$ is True and $q$ is False, therefore the statement is True.

$($b$)9$ is an odd number and $2$ is a negative number.

$($c$)$ If $5>\frac{10}{2}$ then $x^2+x+1=0$ has no real solution.

$($d$)$ This solution is correct if and only if this solution is not wrong.

$(4$pts$)$ Give a logical expression with variables $a,b,$ and $c$ such that

$($a$)$ it is True if a and $b$ are False and $c$ is True; and is False for any other combination of truth values.

$($b$)$ it's truth value does not depend on the value of $b.$

$(10$pts$)$ What is the negation of each of these proposition? Start by translating each English statement into

logic. You must explicitly state what are the atomic propositions. Then, negate the obtained logical statement,

and possibly modify or simplify. After that translate the negated logical statement back into English. Make sure

it sounds natural.

$($a$)$ There's no way I am going to land an interview there.

$($b$)$ Am and Om came to the meeting.

$($c$)$ Umber is a sea cucumber who does not like sea water.

$($d$)$ You must work on a weekend if you need to meet a deadline or catch up on tasks.

$($e$)$ If you eat healthy and exercise regularly you will boost your immune system.