Questions about DFA with Sigma* exiting arrow and RE

Question

Assume Sigma* contains all english alphabet chars. Then in my DFA, I have an exiting arrow of Sigma* and another exiting arrow of "a"(symbol from the alphabet) from one state. Will this be a valid DFA? Since the execution can go two way when seeing an input of "a".. I am not sure.

In Regular Expressions, without writing out all 48 english alphabet chars, is it possible to exclude just some chars from Sigma*? I tried to google this but it just come up with coding stuffs.

Thank you!

Answer 1

If the alphabet is $\Sigma$, then $\Sigma\setminus\{a\}$ is the set of all letters in the alphabet except $a$. This would be a reasonable label for the non-$a$ arrow. If you wanted to exclude both $a$ and $b$, say, you could similarly write $\Sigma\setminus\{a,b\}$. In general $A\setminus B$ is the set of things that are in $A$ but not $B$.