The Closure Property of Positive Real Numbers Under Addition

The closure property under a certain operation tells us that if we the operations between the two members of the set, the result is still a member of that set. In this post, we show that the set of positive real numbers is closed under addition. That is, if we add two real numbers $latex a>0$ and $latex b>0$, then the result $latex a + b>0$.

On the contrary, some sets are not closed under certain operations. For example, the set of irrational numbers are not closed under additionsince

$latex \sqrt{2} + -\sqrt{2} = 0$.

As we can see, both $latex \sqrt{2}$ and $latex – \sqrt{2}$ are irrational numbers, but $latex 0$ is rational.

Theorem

If $latex a>0$ and $latex b>0$, then $latex a + b > 0$ .

Proof

If $latex a > 0$, then adding $latex b$ to both sides, we have $latex a + b > 0 + b$. Now  since $latex a + b > 0$, so by Transitivity Axiom, $latex a + b > 0$.