Question

The set {-1, 0, 1} is not a multiplicative group because of the failure of:
A) Closure law
B) Associative law
C) Identity law
D) Inverse law

Answer 1

Hint: For the given set, we can check the mentioned properties for the operation of multiplication and see what law does not follow, this law will be the reason of failure for the given set to not be a multiplicative group.

Complete step-by-step answer:
We will check all the given properties for the operation of multiplication as this is not a multiplication group.
Let the given set be R = {-1, 0, 1}
A) Closure law: When the two elements are multiplied, the resultant value should belong to the given set.
$\Rightarrow$ -1 $ \times $ 0 = 0 and 0 $ \in $ R
$\Rightarrow$ -1 $ \times $ 1 = 0 and -1 $ \in $ R
$\Rightarrow$ 1 $ \times $ 0 = 0 and 0 $ \in $ R
Thus, the closure law is followed.
B) Associative law: Condition for associative law is –
(a $ \times $ b) $ \times $ c = a $ \times $ (b $ \times $ c)
Here,
a = -1, b = 0, c = 1
Substituting and checking:
$\Rightarrow$ (-1 $ \times $ 0) $ \times $ 1 = -1 $ \times $ (0 $ \times $ 1)
$\Rightarrow$ 0 $ \times $ 1 = -1 $ \times $ 0
0 = 0
Thus, the associative law is followed.
C) Identity law: Presence of 1 in the set, which is true, thus this law is also followed.
D) Inverse law : Inverse of set members should be present within the set.
$\Rightarrow$ -1 🡪 $ \dfrac{1}{{ - 1}} $ = -1 and -1 $ \in $ R
$\Rightarrow$ 0 🡪 $ \dfrac{1}{0} = \infty $ and $ \infty $ does not belong to the set R
Thus, the inverse law is not followed
Therefore, the set {-1, 0, 1} is not a multiplicative group because of the failure of inverse law and the correct option is D)

So, the correct answer is “Option D”.

Note: $ ' \in ' $ symbol means ‘belongs to’.
We use sets to represent a collection of elements in an orderly manner. We generally use following symbols for these common sets:
N: Natural numbers
Z: Integers
Q: Rational numbers
R: Real numbers