Strictly Positive Real Numbers are not Closed under Subtraction
Jump to navigation
Jump to search
Theorem
The set $\R_{>0}$ of strictly positive real numbers is not closed under subtraction.
Proof
Let $a = 1$ and $b = 2$.
Then:
- $a - b = -1$
but $-1$ is not a (strictly) positive real number.
$\blacksquare$
Sources
- 1973: C.R.J. Clapham: Introduction to Mathematical Analysis ... (previous) ... (next): Chapter $1$: Axioms for the Real Numbers: $2$. Fields: Example $1$