Show that each of these conditional statements is a tautology by using truth tables.

a$)[not{p}^{{?}^{?}}(pvvq)]q$

b$)[(pq{)}^{{?}^{?}}(qr)](pr)$

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

d$)[(pvvq{)}^{{?}^{?}}(pr{)}^{{?}^{?}}(qr)]r$

Let $C(x)$ be the statement $"x$ has a cat," let $D(x)$ be the statement $"x$ has a dog," and let $F(x)$ be the statement $"x$ has a ferret." Express each of these statements in terms of $C(x),D(x),F(x),$ quantifiers, and logical connectives.

Let the domain consist of all students in your class.

a$)$ A student in your class has a cat, a dog, and a ferret.

b$)$ All students in your class have a cat, a dog, or a ferret.

c$)$ Some student in your class has a cat and a ferret, but not a dog.

d$)$ No student in your class has a cat, a dog, and a ferret.

e$)$ For each of the three animals, cats, dogs, and ferrets, there is a student in your class who has this animal as a pet.

Translate each of these nested quantifications into an English statement that expresses a mathematical fact. The domain in each case consists of all real numbers.

a$))=(y$

b

c$))))(0)((0)>((0$

d$)))(0)(0)((0$

Use rules of inference to show that the hypotheses $"$If it does not rain or if it is not foggy, then the sailing race will be held and the lifesaving demonstration will go on$,""$If the sailing race is held, then the trophy will be awarded," and "The trophy was not awarded" imply the conclusion $"$It rained."

Prove that if $n$ is an integer and $3n+2$ is even, then $n$ is even using

a$)$ a proof by contraposition.

b$)$ a proof by contradiction.