For the following two questions, prove that the languages are regular by showing how they can be

constructed using the operations from the definition of what a regular language is$.$ For example

Theorem $1.$ The language of all even$-$length strings over the alphabet $\backslash$Sigma $=\{$a$\}$ is regular.

Proof: Let La $=\{$a$\}$ which is regular because this is a base case of the definition of a regular

language.

Let Laa $=$ LaLa $=\{$aa$\},$ which is regular because the concatenation of two regular languages

is regular, by definition.

Let L $=$ L$$

aa$,$ which is also regular because the Kleene$-*$ of a regular language is also regular,

by definition.

But, L is the set of all even$-$length strings over $\backslash$Sigma $.$ Hence, this completes the proof.

$($a$)[5$ marks$]$ The language L of binary strings that either start and end with $0$ or start and end

with $1.$