Binary Operations - Group Theory(Undergraduate Level)

Question

Show that ”+” is not a binary operation on R \ Q by giving an explicit example.

Answer 1

Hi Ashley,

In order for addition to be a binary operation on R \ Q, we must verify that the set R \ Q is closed under addition. That is, if we add any two numbers from R \ Q, the resulting sum must also be in R \ Q.

R \ Q describes the set of real numbers excluding the set of rational numbers, so R \ Q represents the set of irrational numbers.

We see that π and -π both belong to the set R \ Q since they are both irrational. However, if you add these two numbers, the resulting sum is 0, which is rational. This example proves that R \ Q is not closed under addition, so "+" cannot be a binary operation on this set.