Ordering is Preserved on Integers by Addition
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Theorem
The usual ordering on the integers is preserved by the operation of addition:
- $\forall a, b, c, d, \in \Z: a \le b, c \le d \implies a + c \le b + d$
Proof
Recall that Integers form Ordered Integral Domain.
Then from Relation Induced by Strict Positivity Property is Compatible with Addition:
- $\forall x, y, z \in \Z: x \le y \implies \paren {x + z} \le \paren {y + z}$
- $\forall x, y, z \in \Z: x \le y \implies \paren {z + x} \le \paren {z + y}$
So:
\(\ds a\) | \(\le\) | \(\ds b\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds a + c\) | \(\le\) | \(\ds b + c\) | Relation Induced by Strict Positivity Property is Compatible with Addition |
\(\ds c\) | \(\le\) | \(\ds d\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds b + c\) | \(\le\) | \(\ds b + d\) | Relation Induced by Strict Positivity Property is Compatible with Addition |
Finally:
\(\ds a + c\) | \(\le\) | \(\ds b + c\) | ||||||||||||
\(\ds b + c\) | \(\le\) | \(\ds b + d\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds a + c\) | \(\le\) | \(\ds b + d\) | Definition of Ordering |
$\blacksquare$
Sources
- 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): Chapter $2$: Integers and natural numbers: $\S 2.1$: The integers: $\mathbf Z. \, 7$
- 1994: H.E. Rose: A Course in Number Theory (2nd ed.) ... (previous) ... (next): $1$ Divisibility: $1.1$ The Euclidean algorithm and unique factorization