Let * be a binary operation on R defined by a * b = ab + 1. Then,… - Vidyadip

MCQ

Let * be a binary operation on R defined by a * b = ab + 1. Then, * is:

✓
Commutative but not associative.
B
Associative but not commutative.
C
Neither commutative nor associative.
D
Both commutative and associative.

✓

Answer

Correct option: A.

Commutative but not associative.

Commutativity:

Let $\text{a, b}\in\text{R}$

a * b = ab + 1

= ba + 1

= b * a

Therefore,

a * b = b * a, $\forall\text{ a, b}\in\text{R}$

Therefore, * is commutative on R.

Associativity:

Let $\text{ a, b, c}\in\text{R}$

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

= a(bc + 1) + 1

= abc + a + 1

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

= (ab + 1)c + 1

= abc + c + 1

$\therefore$ a * (b * c) $\neq$ (a * b) * c

For example: a = 1, b = 2 and c = 3 [which belong to R]

Now,

1 * (2 * 3) = 1 * (6 + 1)

= 1 * 7

= 7 + 1

= 8

(1 * 2) * 3 = (2 + 1) * 3

= 3 * 3

= 9 + 1

= 10

⇒ 1 * (2 * 3) $\neq$ (1 * 2) * 3

Therefore, $\exists$ a = 1, b = 2 and c = 3 which belong to R such that

a * (b * c) $\neq$ (a * b) * c

Hence, * is not associative on R.

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