Question $1$

Consider ternary $($base$-3)$ numbers over the digits $=\{0,1,2\}.$ Assume that numbers with superfluous leading zeros are allowed. Give a regular expression for each of the following languages:

Numbers where all the digits are the same. For example, $00,1,22,$ and $2222$ are valid, but $010,$ and $122$ are invalid.

Numbers that start and end with the same digit. For example, $1,11$ and $121$ are valid, but $01$ and $122$ are invalid.

Numbers that do not contain consecutive repeated digits. For example, $10$ and $121$ are valid, but $00$ and $122$ are invalid.

Question $2$

Give a DFA for each of the languages in Question $1.$