Construct a regular expression defining each of the

following languages over the alphabet $=\{a,b\}$ :

a$)$ The language of all words that do not begin with ba$.(3$

Points$)$

b$)$ The language of all words in which the total number of

b$\text{'}$s is divisible by $3$ no matter how they are distributed,

such as bbabbaabab. $(3$ Points$)$

c$)$ All words that contain exactly $2$ b$\text{'}$s or exactly $3$ b$\text{'}$s$,$ not

more. $(3$ Points$)$

d$)$ All words in which a appears tripled, if at all. This means

that every clump of $a^{\text{'}}s$ contains $3$ or $6$ or $9$ or $12dots{a}^{\text{'}}s.$

$(3$ Points$)$

e$)$ All words that contain at least one of the strings $S_1,S_2,$

$S_3,$ Or $S_4.(3$ Points$)$