$1$ An Function on Languages

Consider a function, called ADD$0,$ which converts a given input language to a new output language such that for each string $w$ in the input language, there is a corresponding one in the output language that starts with the same characters as $w$ but has an extra $0$ at the end. Formally:

ADD$0(L)=\{w0|$ where $wis$ a string inL$\}$

If the alphabet for a language $L$ is $,$ then the alphabet for ADD$0(L)$ is $0.$

Consider the following language:

$A=\{w|wis$ a three digit number starting with $4\}$

The alphabet for the language is $=\{0,2,4\}.$ Prove that $A$ is a regular language.

$2.$ What is the language ADD$0(A)?($answer must be a set$)$

$3.$ Prove that ADD$0(A)$ is a regular language