Question

Examine whether the operation *defined on R by a * b = ab + 1 is (i) a binary or not. (ii) if a binary operation, is it associative or not?

Answer 1

The given operation is a * b = ab + 1

If any operation is a binary operation then it must follow the closure property.

Let a∈ R, b∈ R
then a*b∈ R
also ab +1 ∈ R
i.e. a *b ∈ R
so * on R satisfies the closure property

Now if this binary operation satisfies associative law then
(a * b) * c = a * (b * c)
(a * b) * c = (ab + 1) * c
= (ab +1) c + 1
= abc + c + 1

a * (b * c) = a * (bc + 1)
= a(bc + 1) + 1
= abc + a + 1

∴ (a * b) * c ≠ a * (b * c)

i.e., * operation does not follow associative law.