Let $\Sigma = \set{0, 1, +, =}$ and

$$ADD = \setbuilder{x=y+z}{x,y,z \in \set{0, 1}^* \text{ and the equation is 'correct' w.r.t. binary arithmetic}}$$

This language is not regular.

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Proof: Suppose $ADD$ were regular and let $p$ be the pumping length. Take $z$ to be the string $1^{2p} = 1^p 0^p + 1^p$, noting that $z \in ADD$ and $|z| \geq p$. Then let $z = uvw$ such that $|uv| \leq p$ and $|v| \geq 1$.

We see that $u$ and $v$ consist entirely of $1$'s, that is, $v = 1^k$ for some $k \geq 1$. But $u v^0 w$ is $1^{(2p - k)} = 1^p 0^p + 1^p$. Looking at the right-hand side, the sum must have $1$ in the $2p$'th spot, so this equation cannot be correct, and this string is not in $ADD$, contradicting the pumping lemma. So $ADD$ is not regular.