Integer Multiples Greater than Positive Integer Closed under Addition
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Theorem
Let $n \Z$ be the set of integer multiples of $n$.
Let $p \in \Z: p \ge 0$ be a positive integer.
Let $S \subseteq n \Z$ be defined as:
- $S := \set {x \in n \Z: x > p}$
that is, the set of integer multiples of $n$ greater than $p$.
Then the algebraic structure $\struct {S, +}$ is closed under addition.
Proof
Let $x, y \in S$.
From Integer Multiples Closed under Addition, $x + y \in n \Z$.
As $x, y > p$ we have that:
- $\exists r \in \Z_{>0}: x = p + r$
- $\exists s \in \Z_{>0}: y = p + s$
Thus it follows that:
\(\ds x + y\) | \(=\) | \(\ds \paren {p + r} + \paren {p + s}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2 p + r + s\) | ||||||||||||
\(\ds \) | \(>\) | \(\ds p\) | as $r, s > 0$ |
So $x + y > p$ and $x + y \in n \Z$.
Hence by definition $x + y \in S$, and so $S$ is closed under addition.
$\blacksquare$
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 8$: Compositions Induced on Subsets: Example $8.1$