TITLE: What phenomenon is this? $(2\Bbb{Z} + 1)\cup 3\Bbb{Z} = 2\Bbb{Z} \cup 3\Bbb{Z} + 3$
QUESTION [6 upvotes]: $(2\Bbb{Z} + 1)\cup 3\Bbb{Z} = 2\Bbb{Z} \cup 3\Bbb{Z} + 3$
Proof:
$$
\begin{align*}
2\Bbb{Z} &= \bullet \circ \bullet \circ \bullet \circ \bullet \circ \dots \\
3\Bbb{Z} &= \bullet \circ \circ \bullet \circ \circ \bullet \circ \dots \\
2\Bbb{Z}+1 &= \circ \bullet \circ \bullet \circ \bullet \circ \bullet \dots \\
2\Bbb{Z} \cup 3\Bbb{Z} &= \bullet \circ \bullet \bullet \bullet \circ \bullet \circ \dots \\
(2\Bbb{Z} + 1)\cup 3\Bbb{Z} &= \bullet \bullet \circ \bullet \circ\bullet\bullet\bullet \dots
\end{align*}
$$
Shift the last one by $\pm 3$. Conclude.  That's a visual proof.  But what is the underlying algebra?

REPLY [2 votes]: It's a compact form of set-builder notation, relying on metaphor to overload the multiplication and addition operators.
$\Bbb Z$ is the set of integers.   And we can define, $a\Bbb Z{+b}$ , to mean, $\{a x + b\mid x\in \Bbb Z\}$ in set-builder notation (or, more formally, $\{y \mid \exists x\in \Bbb Z: y=ax+b\}$ ), where $a,b$ are constants (and usually integers).
Thus this is the set obtained by multiplying each integer by $a$ then adding $b$.   Otherwise known as a scale and shift operation, respectively.
Using this notation we can derive rules, such as distribution: $(a\Bbb Z\cup b\Bbb Z)+c=(a\Bbb Z+c)\cup(b\Bbb Z+c)$ and periodicity: $a\Bbb Z = a\Bbb Z+a$, and so forth, enabling us to manipulate the symbols, thusly:
$$\begin{align}(2\Bbb Z+1)\cup 3\Bbb Z & = (2\Bbb Z+2+1)\cup(3\Bbb Z + 3) 
\\ & = (2\Bbb Z\cup 3 \Bbb Z)+3 \\ \Box\end{align}$$