# How do you proof that for a,binRR, a < b<=>b > a?

Dec 18, 2015

The result generally follows directly from the definition. How the proof is stated will depend on how the definition is phrased. For example let's work with the following definitions:

For $x , y \in \mathbb{R}$ we say

• $x$ is less than $y$ (denoted $x < y$) if $x - y = - k$ for some positive number $k$.

• $x$ is greater than $y$ (denoted $x > y$) if $x - y = k$ for some positive number $k$.

Claim: For $a , b \in \mathbb{R}$, $a < b \iff b > a$

Proof:
$a < b \iff a - b = - k$ for some positive number $k$

$\iff - 1 \left(a - b\right) = - 1 \left(- k\right)$

$\iff b - a = k$

$\iff b - a > 0$

As each step is reversible, this concludes the proof" "▄