$(2Bbb{Z} + 1)cup 3Bbb{Z} = 2Bbb{Z} cup 3Bbb{Z} + 3$
Proof:
$$
begin{align*}
2Bbb{Z} &= bullet circ bullet circ bullet circ bullet circ dots \
3Bbb{Z} &= bullet circ circ bullet circ circ bullet circ dots \
2Bbb{Z}+1 &= circ bullet circ bullet circ bullet circ bullet dots \
2Bbb{Z} cup 3Bbb{Z} &= bullet circ bullet bullet bullet circ bullet circ dots \
(2Bbb{Z} + 1)cup 3Bbb{Z} &= bullet bullet circ bullet circbulletbulletbullet dots
end{align*}
$$
Shift the last one by $pm 3$. Conclude. That’s a visual proof. But what is the underlying algebra?