GATE 1995
March 17, 2025GATE 1997
March 18, 2025GATE 1997
| Question 16 |
Let (Z,*) be an algebraic structure, where Z is the set of integers and the operation * is defined by n*m = maximum (n,m). which of the following statements is true for (Z,*)?
| (Z,*) is a monoid | |
| (Z,*) is an Abelian group | |
| (Z,*) is a group | |
| None of the above |
Monoid – Closed, Associative and has an identity
Group – Monoid with inverse
Abelian group – Group with commutative property
Go through with given:
Closure: Yes.
(m*n = max(m,n)) output is either m or n whichever is maximum since m,n belongs z. The result of the binary operation also belongs to z. So given is satisfying closure property.
Associative: Yes.
The output is max among the elements and it is associative.
Identity: No.
We don’t have single unique element for all the elements which is less than all the elements.
Given one is semigroup only.
Monoid – Closed, Associative and has an identity
Group – Monoid with inverse
Abelian group – Group with commutative property
Go through with given:
Closure: Yes.
(m*n = max(m,n)) output is either m or n whichever is maximum since m,n belongs z. The result of the binary operation also belongs to z. So given is satisfying closure property.
Associative: Yes.
The output is max among the elements and it is associative.
Identity: No.
We don’t have single unique element for all the elements which is less than all the elements.
Given one is semigroup only.