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$]$Given $R=b^{**}(b{a}^{+}{a}^{+}{)}^{+}{b}^{**}$ and $S=(a^{**}b{a}^{+}b{)}^{**}$

a$)$ Give an example of a string that is neither in the language of $R$ nor in $S.[2$ marks$]$

b$)$ Give an example of a string that is in the language of $S$ but not $R.[2$ marks$]$

c$)$ Give an example of a string that is in the language of $R$ but not $S.[2$ marks$]$

d$)$ Give an example of a string that is in the language of $R$ and $S.[2$ marks$]$

e$)$ Design a regular expression that accepts the language of all binary strings with no

occurrences of $001.[3$ marks$]$

f$)$ Design a regular expression that accepts the language of all binary strings which their length

is odd with no occurrences of $11.[3$ marks$]$