Extended transition function for DFA

From CS2800 wiki

Intuitively, when a DFA processes the empty string, it doesn't do anything: if it started in state [math]q [/math], then it stays in state [math]q [/math].

To process the string [math]xa [/math], the DFA would first process the substring [math]x [/math], and then take one more step with the character [math]a [/math] (using the automaton's single step transition function).

This intuition leads to the following definition:

Given an DFA [math]M [/math], we define the extended transition function [math]\href{/cs2800/wiki/index.php/Delta_hat}{\hat{\delta}}:\href{/cs2800/wiki/index.php?title=Q&action=edit&redlink=1}{Q} \href{/cs2800/wiki/index.php/%5Ctimes}{\times} \href{/cs2800/wiki/index.php?title=%5CSigma%5E*&action=edit&redlink=1}{\Sigma^*} \href{/cs2800/wiki/index.php/%5Cto}{\to} \href{/cs2800/wiki/index.php?title=Q&action=edit&redlink=1}{Q} [/math] inductively by [math]\href{/cs2800/wiki/index.php/Delta_hat}{\hat{\delta}}(q,\href{/cs2800/wiki/index.php?title=%5Cvarepsilon&action=edit&redlink=1}{\varepsilon})=q [/math], and [math]\href{/cs2800/wiki/index.php/Delta_hat}{\hat{\delta}}(q,xa)=\href{/cs2800/wiki/index.php/%5Cdelta}{\delta}(\href{/cs2800/wiki/index.php/Delta_hat}{\hat{\delta}}(q,x),a) [/math].