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,
$1^ * (2^ * 3) = 1^ * (6 + 1)$
$= 1^ * 7$
$= 7 + 1$
$= 8$
$(1^ * 2)^ * 3 = (2 + 1)^ * 3$
$= 3^ * 3$
$= 9 + 1$
$= 10$
$\Rightarrow 1^ * (2^ * 3) \neq (1^ * 2)^ * 3$
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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