Congruence (Number Theory) is Congruence Relation
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Theorem
Congruence modulo $m$ is a congruence relation on $\struct {\Z, +}$.
Proof
Suppose $a \equiv b \bmod m$ and $c \equiv d \bmod m$.
Then by the definition of congruence there exists $k, k' \in \Z$ such that:
- $\paren {a - b} = k m$
- $\paren {c - d} = k' m$
Hence:
- $\paren {a - b} + \paren {c - d} = k m + k' m$
Using the properties of the integers:
- $\paren {a + c} - \paren {b + d} = m \paren {k + k'}$
Hence $\paren {a + c} \equiv \paren {b + d} \bmod m$ and congruence modulo $m$ is a congruence relation.
$\blacksquare$
Sources
- 1974: Thomas W. Hungerford: Algebra ... (previous) ... (next): $\text{I}$: Groups: $\S 1$: Semigroups, Monoids and Groups