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$(\text{Q}, \ast)$ is an algebraic structure where $\text{Q}$ represents rational numbers and $\ast$ denotes multiplication. Which one of the following statements is true?

  1. $\text{Q}$ is an abelian group.
  2. $\text{Q}$ is a group but not abelian.
  3. $\text{Q}$ is a semigroup but not a monoid.
  4. $\text{Q}$ is monoid but not group.
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Answer is  D

1)closure: $\forall a,b \in Q , a*b \in Q$, so closure Satisfied.

2)Associativity : All closed groups under multiplication satisfies Associativity,so associativity satisfied.

​​​​​​​3)e=1 , identity element exists.

4)for $0 \in Q$ , inverse does not exist.

    So, $(Q,*)$ is a monoid

 

Answer:

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