Here is a language definition:

$$

$$

$$

$$

$$

$=$

$\{$

$$

$$

$$

is a three$-$digit string that represents a grad$-$level UMB course number

$\}$

L

grad

$$

$=\{$w$$w is a three$-$digit string that represents a grad$-$level UMB course number$\}$

You may assume that strings in the language are drawn from alphabet

$\{$

$0$

$,$

$2$

$,$

$4$

$,$

$6$

$\}$

$\{0,2,4,6\}.$

Come up with $3$ strings in the language.

Come up with $3$ strings not in the language.

For this problem, for parts $1$ and $2,$ you are allowed to post examples to Piazza for clarification if you are unsure.

Prove that the language is a regular language.

Important: Before attempting the proof, make sure to understand what this problem is asking. This means you must know the precise definitions of the terms used in this problem before you can begin solving this question, e$.$g$.,$ make sure you know the precise formal definition of: string, language, regular language, etc. Further, you need to know how to read the set $($builder$)$ notation above and what it means for something to be in and not in a set.

Your proof must be in the form of a Statements and Justifications table, as presented in class.

If your answer includes designing a DFA, you may submit a state diagram instead of a formal description of the machine.

To "prove" that a DFA recognizes a language, you may use as justification a table showing three strings that are both accepted by the DFA and in the language, and three strings that the DFA rejects and are also not in the language, as shown in lecture. The strings should ideally "test" as many different transition paths through the DFA as possible. $($This should also tell you whether the strings you came up with in parts $1$ and $2$ are "good" examples or not.$)$