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B> Not a group but monoid because the inverse element does not exist as they will not be integers

eg: the multiplicative inverse of 3 will be 1/3 which donot closed under positive integer 

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The set of all positive integers ($\mathbb{Z}^{+}$)  =   { 1, 2, 3, 4, 5, 6…...}

A set A under a binary operation * can be called as a group G iff it satisfies

               Closure property

               Associativity

               Identity property

               Inverse property

 

Here the set contains an identity element which is 1….But the set doesn’t hold inverse property 

Eg: for element 2, we cannot find an inverse such that  when we multiply any element of the set A (positive

integers) we get the identity element 1...

Hence it cannot be a Group…

It is a Monoid (A,*) since the set A under multiplication satisfies Closure, Associative and Identity property….

Since it also satisfies Commutative property , it can be called a Abelian Monoid...

The best option would be B...

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Correct Answer is: a) not a group

The set of positive integers under the operation of ordinary multiplication forms a monoid.

A monoid is a set equipped with an associative binary operation and an identity element. In this case, the set of positive integers has the multiplication operation, which is associative (a * b) * c = a * (b * c), and the identity element is 1 (since any positive integer multiplied by 1 is the integer itself).

However, it is not a group because not all elements have inverses under multiplication. For example, there is no positive integer x such that 2 * x = 1.

Therefore, the correct answer is:

a) not a group

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