Decidability of Unary Languages / One-to-One Mapping
Instead of mapping the string $x\in\{0,1\}^*$ to $1^{\mathrm{bin}(x)}$ (where $\mathrm{bin}(x)$ is the number denoted by interpreting&nbsp;$x$ as a string in binary, map it to $1^{\mathrm{bin}(1x)}$. Now every string in $\{0,1\}^*$ maps to a unique number of $1$s.