Real Number Ordering is Compatible with Addition

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Theorem

$\forall a, b, c \in \R: a < b \implies a + c < b + c$

where $\R$ is the set of real numbers.


Proof

From Real Numbers form Ordered Integral Domain, $\struct {\R, +, \times, \le}$ forms an ordered integral domain.

By definition of ordered integral domain, the usual ordering $\le$ is compatible with ring addition.

$\blacksquare$


Examples

$15 + 3$ Greater than $12 + 3$

We have that:

$15 > 12$

so by Real Number Ordering is Compatible with Addition:

$15 + 3 > 12 + 3$

That is:

$18 > 15$


$15 - 3$ Greater than $12 - 3$

We have that:

$15 > 12$

so by Real Number Ordering is Compatible with Addition:

$15 - 3 > 12 - 3$

That is:

$15 > 12$


Sources