Modulo Subtraction is Well-Defined
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Corollary to Modulo Addition is Well-Defined
Let $m \in \Z$ be an integer.
Let $\Z_m$ be the set of integers modulo $m$.
The modulo subtraction operation on $\Z_m$, defined by the rule:
- $\eqclass a m -_m \eqclass b m = \eqclass {a - b} m$
is a well-defined operation.
That is:
- If $a \equiv b \pmod m$ and $x \equiv y \pmod m$, then $a - x \equiv b - y \pmod m$.
Proof
We have:
\(\ds \eqclass a m -_m \eqclass b m\) | \(=\) | \(\ds \eqclass {a - b} m\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \eqclass {a + \paren {-b} } m\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \eqclass a m +_m \eqclass {-b} m\) |
The result follows from the fact that Modulo Addition is Well-Defined for all integers.
$\blacksquare$
Examples
Modulo Subtraction: $19 - 6 \equiv 11 - 2 \pmod 4$
We have:
\(\ds 19\) | \(\equiv\) | \(\ds 11\) | \(\ds \pmod 4\) | |||||||||||
\(\ds 6\) | \(\equiv\) | \(\ds 2\) | \(\ds \pmod 4\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds 19 - 6 = 13\) | \(\equiv\) | \(\ds 11 - 2 = 9\) | \(\ds \pmod 4\) |
Sources
- 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 2.6$. Algebra of congruences: $\text{(ii)}$
- 1971: George E. Andrews: Number Theory ... (previous) ... (next): $\text {4-1}$ Basic Properties of Congruences: Theorem $\text {4-2}$
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 14.3 \ \text {(ii)}$: Congruence modulo $m$ ($m \in \N$)
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.4$: Integer Functions and Elementary Number Theory: Law $\text{A}$
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): congruence