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 and , then the result .

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

.

As we can see, both and are irrational numbers, but is rational.

**Theorem**

If and , then .

**Proof**

If , then adding to both sides, we have . Now since , so by Transitivity Axiom, .

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