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?

*: a*b=ab+1

(i) Given operation is a*b=ab+1

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

Let a∈R and b∈R

Then ab∈R

Also ab+1 ∈R

So, a*b ∈R

So * satisfies the closure property.

Since * is defined for all a, b ∈ R, therefore * is a binary operation.

(ii) For * to be associative, (a*b) *c=a*(b*c)

(a*b) *c=(ab+1) *c

=abc+c + 1

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

= a(bc+1) + 1

=abc+a+1

Since (a*b) *c≠ a*(b*c), therefore * is not associative.