$1)$ Answer the following questions. Explain your reasoning: a$)$ Without using the truth table, show that the following statement is a tautology: $[3$ marks$]$

$((($R$->$S$)->$R$)->$R$)$

b$)$ Without using the truth table, show that the following statement is true: $[3$ marks$]$

$(($P$$Q$)($P$$Q$))($P$$Q$)$

c$)$ Negate the following statement: $[3$ marks$]$

$$x: P$($x$)[$ x:$($Q$($x$)()]$

d$)$ For the following statement, write down a logically equivalent statement which contains no operators other than and $->$: $[3$ marks$]$

P $$ Q $$ R

Answer the following questions. Explain your reasoning:

a$)$ Without using the truth table, prove if the following statements are tautologies or not.

Show your reasoning. $[4$ marks$]$

i$),pvv(pq)vvnotq$

ii$),((Pq{)}^{{?}^{?}}((q^{{?}^{?}}r)s){)}^{{?}^{?}}(r(Ps))$

b$)$ Without using the truth table, show that the following statement is true: $[2$ marks$]$

not$(EEx[P(x{)}^{{?}^{?}}Q(x)])-=$AAx$[P(x)$notQ$(x)]$

c$)$ Negate the following statement: $[2$ marks$]$

AAx:$P(x{)}^{{?}^{?}}[EEx$:$(Q(x{)}^{{?}^{?}}$notR$(x))]$

d$)$ For the following statement, write down a logically equivalent statement which contains

no operators other than not and $V$ : $[2$ marks$]$

$P^{{?}^{?}}(QR)$