$($a$)$ Determine a regular expression for the language $L$ over the alphabet $\{a,b\}$ that ends

with a letter that is different from the letter that it starts with.

For example, $ab,$ababab and aaaaabaaaabbbbbb are words in $L$ but aaa,baabbaab,

aaaaabaaaaaa and abaa are not.

$($b$)$ Design a deterministic finite automaton $($DFA$)$ that will recognise all of the words in

as defined above.

$($c$)$ Use Theorem $21$ to develop a context$-$free grammar $($CFG$)$ for the language $L.$

$($d$)$ Convert the following CFG to Chomsky Normal Form $($CNF$)$:

$S$bYaZ$|axaY$

$x$baY$|abZ$

$Yax|$

$Zbx|$