A string of $0$ s and $1s$ is to be processed and converted to an even$-$parity string by adding a parity bit to the end of the string $($defined in section $3\mathrm{.}1).$ The parity bit is initially $0.$ When a $0$ character is processed, the parity bit remains unchanged. When a $1$ character is processed, the parity bit is switched from $0$ to $1$ or $1$ to $0.$ Prove that the number of $1s$ in the final string, including the parity bit, is always even. Consider all the various cases and SHOW ALL YOUR STEPS. Carefully explain your proof and all your reasonings.