Question

In the group of non-zero rational numbers under the binary operation * given by a*b=ab/5 the identity element and the inverse of 8 are respectively
(
A)5 and 5/8
(B)5 and 25/8
(C)5 and 8/25
(D)None of these

Answer 1

Answer:

A)5 and 5/8 this is my answer from bushra 35

Answer 2

Answer:

The identity element is 5 and inverse of 8 is 25/8.

Step-by-step explanation:

Binary operation:

• A binary operation is a rule for combining two elements in a set to produce another element in the same set.
• In other words, If  A  be a non-empty set, and  $*$  is a binary operation on  A, the  $a* b$  is defined for all  $a,b \in A$.  

Identity element:

• If  $*$  is a binary operation on  A, an element $e \in A$ is an identity element of A with respect to the operation, if
• $\forall a \in A, a *e = e * a = a$

Inverse element:

• If  $*$  is a binary operation on  A, $a \in A$ and identity element $e \in A$, then a is invertible with respect to the operation, if there exists $b \in A$ so that
• $a *b = b * a = e$

Given group of non-zero rational numbers under the binary operation $&$$*$ given by

$a*b=\dfrac{ab}{5}$

(i) Using the definition of identity element,

$a *e = a\implies \dfrac{ae}{5}=a$

              $\implies e=5$

(ii) Given to find the inverse of 8, i.e., $a=8$

Using the definition of inverse element,

$a *b = e\implies \dfrac{8\times b}{5}=5$

              $\implies b=\dfrac{25}{8}$

Therefore, the identity element is 5 and inverse of 8 is 25/8.

So, the correct answer is option B.

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