$.$ Prove that language A $=$ fambn j n $>$ mg over alphabet fa; bg is not

regular. Make your proof clear and readable. Do not expect the reader

to guess what you are doing. Follow this outline.

$($a$)$ The proof is by contradiction. Suppose that A is regular. That

means there is a DFA M that solves A$.$ Do an experiment on A

by running it on a sequence of strings. What is the sequence of

strings that you run A on$?$

$($b$)$ What can you conclude from the experiment in part $($a$)?$

$4$

$($c$)$ Using the information from part $($b$),$ show that M does not solve

A$.$ That contradicts the assumption that M solves A$.$