Prove L = {a = b ⊕ c | a, b, c ∈{0, 1}*} is not regular
Suppose $L$ were regular. Let $p$ be the pumping constant assumed to exist and consider the string 

$$s = 1^p = 1^p \oplus 0^p$$

Clearly, $|s| \geq p$ so for some strings $x,y,z$ we have $s = xyz$ where


$|xy| \leq p$
$|y| &gt; 0$
$xy^iz \in L$ for all $i \geq 0$


If condition 1 is to hold, $y$ must be a string of $1$s. If condition 2 is to also hold, then $y = 1^k$ for some $k &gt; 0$. But then condition 3 fails, since $xy^0z \notin L$.

For $xy^0z =  1^{p-k} = 1^p \oplus 0^k$, and this identity is false irrespective of whether we require $w_1 \oplus w_2$ to be defined for strings of the same length only, or in general.