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Let $m$ be any fixed positive integer. For each integer $j$, $0\le j \lt m$, let $\Bbb Z_j=\{x \in \Bbb Z\,|\, x-j=km \text{ for some } k \in \Bbb Z\}$.

Then when we consider the below definition, can we denote $\{x \in \Bbb Z\,|\, x-j=km \text{ for some } k \in \Bbb Z\}$ as $\bigcup\limits_{k \in Z}${km+j} or $\overset {m-1}\bigcup\limits_{j=0}${km+j}?

FYI "Definition 6 Let F be an arbitrary family of sets. The union of the sets in F, denoted by $\bigcup\mathscr F$, is the set of all elements that are in A for some $A\in\mathscr F$.​

$\bigcup\limits_{A \in \mathscr F}A$={$x\in U$|x∈A for some $A\in F$}"

If the family $\mathscr F$ is indexed by the set $\Gamma$, the following alternate notation may be used:

$\bigcup\limits_{r\in \Gamma}A_r$={$x\in U$ | $x \in A_r$ for some $r\in \Gamma$}

If the idex set $\Gamma$ is finite, $\Gamma$={1, 2, 3,..., n} for some natural number n, more intuitive notations such as

$\overset {n}\bigcup\limits_{i=1}A_i$ or $A_1 \bigcup A_2 \bigcup \cdots \bigcup A_n$ are often used for $\bigcup_{r \in \Gamma}A_r$.

Source: Set Theory by You-Feng Lin, Shwu-Yeng T. Lin.

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That is correct, although it is customary to denote such a union of singletons as

$$\{km+j|k\in\mathbb Z\}$$

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You also can denote the partition associated with the congruence in a compact form as $$\mathbf Z=\bigcup_{j=0}^{m-1}(j+\mathbf Zm).$$

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  • $\begingroup$ No, it can't be denoted that way. What you have written is equal to $\mathbb Z$, and the set the OP is asking about is not equal to $\mathbb Z$. $\endgroup$ – hmakholm left over Monica Jan 12 '16 at 13:14
  • $\begingroup$ @Henning Malcolm: Right, I should have used it as a way to specify the partition induced by the congruence classes. $\endgroup$ – Bernard Jan 12 '16 at 13:22

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