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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
in Set Theory & Algebra by Veteran (52.3k points)
edited by | 2k views
0
what about a*a = max(a, a)?How it can be an algebric structure if max value of the same elements does not exists. Please correct me if i am wrong
0
If the set was for Natural numbers we could have had 1 as identity element as it is less than any other natural number and we would have got a monoid.

what about max(1,1)=??
0
Hello sushmita

What is restraining you from considering $max(n,n)=n$ ?
0
If set would have been of +ve integers then we've 1 as identity right...?

6 Answers

+24 votes
Best answer

Lets follow our checklist one by one to see what property this algebraic structure follows.

Closure -yes ($m*n=\max(m,n)$) Output is either $m$ or $n$ whichever is maximum and since $m,n$ belongs to $Z$, the result of the binary operation also belongs to $Z$. So closure property is satisfied.

Associative-Yes the output is max among the elements and it is associative

Now for identity, we don't have a single unique element for all the elements which is less than all the elements. ie, $m*e=m \implies \max(m,e) = m$. We can't find a single unique $e$ which is less than all possible integer $m$, such that comparison between the two would always give $m$ itself.

If the set was for Natural numbers we could have had 1 as identity element as it is less than any other natural number and we would have got a monoid.

  • Semi-group - Closed and associative
  • Monoid - Closed, associative and has an identity
  • Group - Monoid with inverse
  • Abelian group- Group with commutative property.


Hence this is just a semigroup- D option.

Ans D)

by Active (3.6k points)
selected by
0
@Gabbar, The smallest element in the group can act as identity element right?

because, Max(smallest element, anything) = anything isn't? Herea smallest integer will act as identity element doesn't?
+2
now tell me which is the smallest integer?? can u tell?? can anyone??
0
-infinity
0
It is like - infinity is answer if - infinity-1 taken in to consideration for max(n,m). On number line go from 0 to left(negative side) you can't give a fix value which can serve the purpose. :)
0
I have doubt i don't know why i am thinking like that ...

* identity is 1

+ identity is 0

so for identity a*e=a should satisfy-

lets take an ex:- 2*1=max(2,1)=2

                           3*1=3

and 1 is also present in set of natural no ?
0

Z is set of integers means { -infinity........-2, -1, 0, 1, 2........infinity}, not like set of natural numbers {1,2,3,4,......infinity}

How can u find the smallest element in Z ???

that's why Z is not a monoid

+1

@ Relax. No need to be angry :p

+4 votes

Given (Z,*) is an algebraic structure. Hence it is closed.

Now, to become SEMI-GROUP: It should be associative

(x*y)*z = max(max(x,y),z) = max(x,y,z)

x*(y*z) = max(x,max(y,z)) = max(x,y,z)

eg. (1*2)*3 = max((max(1,2),3) = max(2,3) = 3

1*(2*3) = max(1,max(2,3)) = max(1,3) = 3

Hence (x*y)*z = x*(y*z), so ASSOCIATIVE i.e SEMI-GROUP

Now, to become MONOID: It should have an identity element

Let there be such an element and let the identity element be e.

So, as per definition: (a*e) = max(a,e) = a

Nice. But the question is: What if we take e=(a+1) ∀ a ∈ Z

Then, (a*(a+1)) = max(a,(a+1)) = (a+1) and NOT a

So, there can be no such identity(e) element possible.

Hence, its not a MONOID.

So correct choice is: (D) None

by (245 points)
edited by
+4 votes
Closure Property => This is satisifed .Maximum of two integer is one of the integers. So we satisfy closure property.

Associativity => This property also satisfied.

2,3,4 =Max(2(max(3,4)) = 4 = Max(Max(2,3),4) .

Identity => We do not have identity.

Suppose some i is identity.

Then Max(i,anything) should be that anything !

But if I  take i-1. (As we have Z , If i is integer so is i-1)

Max(i,i-1) = i. So i is not identity.

So This is not monoid. & SO not group or abelian group . Ans is D
by Boss (42.1k points)
+3 votes
Hi , in this question does identity element always exist ? because , if I take a set A = {-2,-1,0,1,2}

( 0 can be included as it is integer set ).

so , -2 * e = -2 ( where e is the identity element ) , but if here we can not have any such unique identity element .

So , this is not going to be Monoid .

Please correct me , if I am wrong.
by Active (3.8k points)
+1 vote

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

by Boss (11.7k points)
0 votes

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

by Loyal (7.7k points)
edited by
Answer:

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