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If $A$ and $B$ are subsets of $\mathbb{Z}$, define $AB = \{ab : a ∈ A \wedge b ∈ B\}$.

For each integer $x$, let $[x]$ be the equivalence class of $x$ in $\mathbb{Z}$ with respect to congruence modulo $n$. Then the equation $[x][y] = [xy]$ holds.

I'm having a lot of trouble proving this one. Since $[x][y]$ and $[xy]$ are both sets, we should be able to demonstrate that they are equal by demonstrating that each is a subset of the other. I have had no problem demonstrating that $[x][y]\subseteq[xy]$, but I'm having a significant amount of trouble showing that $[xy]\subseteq[x][y]$. Does anyone have any ideas?

Jay
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1 Answers1

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Hints:

Suppose that $x=k_1n+s$ and $y=k_2n+t$, where $0\le s,t <n$. So $[xy]=[(k_1n+s) \cdot (k_2n+t)]=[s\cdot t]$

Clearly $s \in [x]$ and $t\in [y]$, so, $s\cdot t\in [x][y]$.

Paul
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