Let $\Sigma_{1}$ and $\Sigma_{2}$ be alphabets of sizes $m$ and $a$. What does it mean to say that $f: \Sigma_{1} \rightarrow \Sigma_{2}^{*}$ is a decipherable code? State the inequalities of Kraft and Gibbs, and deduce that if letters are drawn from $\Sigma_{1}$ with probabilities $p_{1}, \ldots, p_{m}$ then the expected word length is at least $H\left(p_{1}, \ldots, p_{m}\right) / \log a$.