Let $\Sigma_1$ and $\Sigma_2$ be two alphabets. A homomorphism is a total function $h$ from $\Sigma_1^*$ to $\Sigma_2^*$ that preserves concatenation. That is, $h$ satisfies
i) $h(\lambda)=\lambda$
ii) $h(u v)=h(u) h(v)$.
a) Let $\mathrm{L}_1 \subseteq \Sigma_1^*$ be a regular language. Show that the set $\left\{h(w) \mid w \in \mathrm{L}_1\right\}$ is regular over $\Sigma_2$. This set is called the homomorphic image of $L_1$ under $h$.
b) Let $\mathrm{L}_2 \subseteq \Sigma_2^*$ be a regular language. Show that the set $\left\{w \in \Sigma_1^* \mid h(w) \in \mathrm{L}_2\right\}$ is regular. This set is called the inverse image of $\mathrm{L}_2$ under $h$.