Let * be a binary operation on R defined by a * b = ab + 1. Then, * is: 1. Commutative but not associative. 2. Associative but not commutative. 3. Neither commutative nor associative. 4. Both commutative and associative.

1. Commutative but not associative. Solution: Commutativity: Let a, b a * b = ab + 1 = ba + 1 = b * a Therefore, a * b = b * a, a, b Therefore, * is commutative on R. Associativity: Let a, b, c 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 a * (b * c) (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) (1 * 2) * 3 Therefore, a = 1, b = 2 and c = 3 which belong to R such that a * (b * c) (a * b) * c Hence, * is not associative on R.

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

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Question

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

Commutative but not associative.
Associative but not commutative.
Neither commutative nor associative.
Both commutative and associative.

✓

Answer

Commutative but not associative.

Solution:
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.