Let * be the binary operation on N defined by a * b = H.C.F. of a and b . Is * commutative? Is * associative? Does there exist identity for this binary operation on N ?
It is given that the binary operation ∗ on N is defined as a ∗ b = H .C .F . ( a , b ) .
Apply the given binary operation on b ∗ a .
b ∗ a = H .C .F . ( b , a ) = H .C .F . ( a , b )
The value of a ∗ b = H .C .F . ( a , b ) is equal to b ∗ a = H .C .F . ( a , b ) . So, it satisfies commutative property.
So, the operation ∗ is commutative.
Consider three variables for associativity, that are a , b and c .
Apply the given binary operation on ( a ∗ b ) ∗ c .
( a ∗ b ) ∗ c = ( H .C .F . ( a , b ) ) ∗ c = H .C .F . ( a , b , c )
Apply the given binary operation on a ∗ ( b ∗ c ) .
a ∗ ( b ∗ c ) = a ∗ ( H .C .F . ( b , c ) ) = H .C .F . ( a , b , c )
The value of ( a ∗ b ) ∗ c is equal to a ∗ ( b ∗ c ) . So, it satisfies the property of associative.
Therefore, the binary operation ∗ is associative.
An element e ∈ N is said to be an identity for the operation ∗ when the case given below is valid,
a * e = a e * a = a
But the relation is not true for any a ∈ N .
Thus, the operation ∗ does not have any identity in N .
It is given that the binary operation
Apply the given binary operation on
The value of
So, the operation
Consider three variables for associativity, that are
Apply the given binary operation on
Apply the given binary operation on
The value of
Therefore, the binary operation
An element
But the relation is not true for any
Thus, the operation
