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Let $\left(Z, *\right)$ be an algebraic structure where $Z$ is the set of integers and the operation $*$ is defined by $n*m = \max(n,m)$. Which of the following statements is true for $\left(Z, *\right)$?

  1. $\left(Z, *\right)$ is a monoid
  2. $\left(Z, *\right)$ is an Abelian group
  3. $\left(Z, *\right)$ is a group
  4. None of the above
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A- For Monoid, semigroup should have identity property

B-For Abelian group, group should have commutative property

C-For Group Monoid should have a inverse property

But there is no property n*m=max(n,m) so 

Option D must be True

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Closure? Yes.

If you take two integers, the maximum out of the two would be one of these two. Means an integer.


Associativity. Yes

Position of the sequence doesn't matter.

Example: Maximum of a and b then c $\equiv$ Maximum of b and c then a


Identity?

We need an e, such that $x*e=x$ $=>$ $max(x,e)=x$

So, e must be the smallest negative integer. Which is $-∞$; which actually is just an abstraction. There's no defined number that represents $-∞$

So, identity doesn't exist.

 

Hence, the given structure is a Semi Group. Option D

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