is closed;

is commutative;

is associative;

has an identity element.

\( * \) is closed     A1

because \(1 + ab \in \mathbb{N}\) (when \(a,b \in \mathbb{N}\))     R1

[2 marks]

consider

\(a * b = 1 + ab = 1 + ba = b * a\)     M1A1

therefore \( * \) is commutative

[2 marks]

EITHER

\(a * (b * c) = a * (1 + bc) = 1 + a(1 + bc){\text{ }}( = 1 + a + abc)\)     A1

\((a * b) * c = (1 + ab) * c = 1 + c(1 + ab){\text{ }}( = 1 + c + abc)\)     A1

(these two expressions are unequal when \(a \ne c\)) so \( * \) is not associative     R1

OR

proof by counter example, for example

\(1 * (2 * 3) = 1 * 7 = 8\)     A1

\((1 * 2) * 3 = 3 * 3 = 10\)     A1

(these two numbers are unequal) so \( * \) is not associative     R1

[3 marks]

let e denote the identity element; so that

\(a * e = 1 + ae = a\) gives \(e = \frac{{a - 1}}{a}\) (where \(a \ne 0\))     M1

then any valid statement such as: \(\frac{{a - 1}}{a} \notin \mathbb{N}\) or e is not unique     R1

there is therefore no identity element     A1

Note: Award the final A1 only if the previous R1 is awarded.

[3 marks]

For the commutative property some candidates began by setting \(a * b = b * a\) . For the identity element some candidates confused \(e * a\) and \(ea\) stating \(ea = a\) . Others found an expression for an inverse element but then neglected to state that it did not belong to the set of natural numbers or that it was not unique.

For the commutative property some candidates began by setting \(a * b = b * a\) . For the identity element some candidates confused \(e * a\) and \(ea\) stating \(ea = a\) . Others found an expression for an inverse element but then neglected to state that it did not belong to the set of natural numbers or that it was not unique.

For the commutative property some candidates began by setting \(a * b = b * a\) . For the identity element some candidates confused \(e * a\) and \(ea\) stating \(ea = a\) . Others found an expression for an inverse element but then neglected to state that it did not belong to the set of natural numbers or that it was not unique.

For the commutative property some candidates began by setting \(a * b = b * a\) . For the identity element some candidates confused \(e * a\) and \(ea\) stating \(ea = a\) . Others found an expression for an inverse element but then neglected to state that it did not belong to the set of natural numbers or that it was not unique.