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$),((Pq{)}^{{?}^{?}}((q^{{?}^{?}}r)s){)}^{{?}^{?}}(r(Ps))$Design a Finite State Automaton $($FSA$)$ that accepts binary numbers greater than $0,$ whose

decimal equivalents are divisible by $5.$ For instance, the FSA should accept inputs like

$\text{'}1010\text{'}($decimal equivalent: $10),$ and $\text{'}101\text{'}($decimal equivalent: $5),$ but it rejects inputs like

$\text{'}1101\text{'}($decimal equivalent: $13)$ and $\text{'}1001\text{'}($decimal equivalent: $9).[4$ marks$]$

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$]$

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