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