Show that {xy : x,y ∈ {a,b}*, |x| = |y|, x ≠ y} is a not a regular language

Question

Actually, I know that there are many examples showing how this is a contex-free language, but I can't find any that show it isn't regular. I would appreciate if I could have a solution step by step for this case Show that {xy : x,y ∈ {a,b}*, |x| = |y|, x ≠ y} is not a regular language

Answer 1

You need to use the concept of pumping lemma.

1. Define a pumping length, $p$

2. Choose a word, $w\in L $ such that $ |w|\geq p$

3. Split $w$ into three constituent parts, $x,y,z$ such that $y\neq\varepsilon$ (but $x$ and $z$ are allowed to $=\varepsilon$), $|xy|\leq p$, and $xy^kz$ is in your language $L$ for any $k \geq0$.

4. Let's leave $p$ undefined as $p$ for this, and take the word $a^p b^p$ which satisfies the rules of the language

5. $|w|=2p\geq p$ therefore the condition in step 2 is met

6. Now we split up $w$. With the rules defined, $y$, the important bit, must be in the first $p$ characters of $w$, meaning that it must be made up entirely of $a$'s, so let's take $x=a$, $y = a^{p-1}$, $z=b^p$ (but it doesn't matter whether $y$ is $a^1$, or $a^p$, and $x=\varepsilon$ or $x=a^{p-1}$, the outcome is the same).

7. Now for the important part - if we pump the string such that $xy^0z$, we get $a(a^{p-1})^0b^p=ab^p$ which you'll observe is $\notin L$.

8. In fact, whatever we use as $k$ to pump $y$, we'll always end up with a scenario where $|x|\neq|y|$, and this would be the same whatever string you used from your language.