(a) Which of the following are regular languages? Justify your answers.

(i) $\left\{w \in\{a, b\}^{*} \mid w\right.$ is a nonempty string of alternating $a$ 's and $b$ 's $\}$.

(ii) $\left\{w a b w \mid w \in\{a, b\}^{*}\right\}$.

(b) Write down a nondeterministic finite-state automaton with $\epsilon$-transitions which accepts the language given by the regular expression $(\mathbf{a}+\mathbf{b})^{*}(\mathbf{b} \mathbf{b}+\mathbf{a}) \mathbf{b}$. Describe in words what this language is.

(c) Is the following language regular? Justify your answer.

$$\left\{w \in\{a, b\}^{*} \mid w \text { does not end in } a b \text { or } b b b\right\}$$