Number of binary operations

The number of binary operation on {1,2,3,......,n} is ..........

The number of binary operation on set {1,2} is .......

Show that the total number of binary operation from set A to A is n^(n^(2)).

Let A= {a+sqrt5b:a,binZ} . Show that usual multiplication of numbers is binary operation on A

On Q , the set of all rational numbers a binary operation * is defined by a*b= (a+b)/2 . Show that * is not associative on Qdot

On Q, the set of all rational numbers a binary operation * is defined by a*b=(a+b)/(2) Show that * is not associative on Q.

On the set R of real numbers, a binary operation o is defined by a0b=a+b+a^2b AA a, b inR . Prove that the operation is neither commutative nor associative.

On the set Q of all rational numbers if a binary operation * is defined by a*b= (a b)/5 , prove that * is associative on Qdot

On the set Q of all rational numbers if a binary operation * is defined by a*b=(ab)/(5), prove that * is associative on Q.

On Q, the set of all rational numbers, a binary operation * is defined by a*b=(a b)/5 for all a , b in Qdot Find the identity element for * in Q. Also, prove that every non-zero element of Q is invertible.