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 a>0 and b>0, then the result 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

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

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

Theorem

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

Proof

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

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