Question

Show that addition, subtraction and multiplication are binary operationson R, but division is not a binary operation on R. Further, show that division is a binaryoperation on the set R∗

of nonzero real numbers.​

Answer 1

Solution + : R × R → R is given by (a, b) → a + b

– : R × R → R is given by (a, b) → a – b

× : R × R → R is given by (a, b) → ab

Since ‘+’, ‘–’ and ‘×’ are functions, they are binary operations on R.But ÷: R × R → R, given by (a, b) →

a/bis not a function and hence not a binaryoperation, as for b = 0, a/bis not defined.However, ÷ : R∗ × R∗ → R∗

, given by (a, b) →a/b is a function and hence abinary operation on R∗

.