Modulo Addition is Well-Defined/Real Modulus

From ProofWiki
Jump to navigation Jump to search

Theorem

Let $z \in \R$ be a real number.

Let:

$a \equiv b \pmod z$

and:

$x \equiv y \pmod z$

where $a, b, x, y \in \R$.


Then:

$a + x \equiv b + y \pmod z$


Proof

\(\ds a\) \(\equiv\) \(\ds b\) \(\ds \pmod z\)
\(\ds c\) \(\equiv\) \(\ds d\) \(\ds \pmod z\)
\(\ds \leadsto \ \ \) \(\ds a \bmod z\) \(=\) \(\ds b \bmod z\) Definition of Congruence
\(\ds x \bmod z\) \(=\) \(\ds y \bmod z\)
\(\ds \leadsto \ \ \) \(\ds \exists k_1 \in \Z: \, \) \(\ds a - b\) \(=\) \(\ds k_1 z\)
\(\ds \exists k_2 \in \Z: \, \) \(\ds x - y\) \(=\) \(\ds k_2 z\)
\(\ds \leadsto \ \ \) \(\ds \paren {a + x} - \paren {b + y}\) \(=\) \(\ds \paren {k_1 + k_2} z\) Definition of Integer Addition
\(\ds \leadsto \ \ \) \(\ds a + x\) \(\equiv\) \(\ds b + y\) \(\ds \pmod z\) Definition of Congruence

$\blacksquare$


Sources

Retrieved from "https://proofwiki.org/w/index.php?title=Modulo_Addition_is_Well-Defined/Real_Modulus&oldid=539250"