Question

A closed set with respect to some binary operation is called semigroup if
(A) * is associative
(B) * is commutative.
(C) * is anti-commutative.
(D) * Identity element exists.

Answer 1

Hint: At first we have to know about semi group –
A semigroup is a pair (S, .) in which S is a non-empty set and ‘.’ Is a binary associative operation on S. i.e the equation $\left( {x.y} \right).z = x.\left( {y.z} \right)$should for all $x,y,z \in s.$
As long or not otherwise stated, we write the semi group operation is multiplication. And we mostly omit it typographically .i.e. write S instead of (S, .), xy instead of x.y, x(y z) instead of x. (y. z) and So on.

Complete step by step solution:
Let the corridor, an algebraic system $\left( {A,*} \right)$, where * is a binary operation on A. Then the system $\left( {A,*} \right)$is said to be semigroup if it satisfier the following properties
(i) The operation * is a closed operation on set A
(ii) The operation * is an associative operation
Example – Consider an algebraic system $\left( {A,*} \right)$, where $A = \left\{ {1,3,5,7.........} \right\}$the set of positive add integers and * is a binary operation evener multiplication, Determine whether $\left( {A,*} \right)$ is a semi-group
closure property: The operation * is closed operation because multiplication of two +ve add integers is a +ve add number
Associative property :
The operation * is an associative operation on set A, Since every $a,b,c \in A$we have
$\left( {a*b} \right)*e = a*\left( {b*c} \right)$
Hence; The algebraic system $\left( {A,*} \right)$ is a semi-group.
So by this discussion, we can see that the binary operation has to be associative.
The correct option is (A)

Note: In mathematics, a semi-group is a non-empty set together with an associative binary operation. Special care of semi-group satisfying additional properties or conditioners. Thus the Clare of commutative semigroup commits all those semi-groups in which the binary operation satisfies the commutative properly that ab = ba for all elements a and b in the semi-group.