Is the binary operation `**` defined by Q by `a**b=(a+b)/2` for all `a,b in Q` commutative and associative?

Let A=NxxN and let ** be a binary operation on A defined by (a,b)**(c,d)=(a+c,b+d) . Show that ** is commutative and associative.

Show that the operation * defined on Q-{1} by a**b=a+b-ab is abinary operation. Is * commutative and associative? What is the identity element wrt *? Which elements possess inverses?

Is the binary operation ** defined on Z (set of integers) by the rule a**b=a-b+ab for all a, bin Z commutative ?

Let * be a binary operation defined by a*b = 3a+4b-2 find 4*5

For the binary operation ** defined below, determine whether ** is commutative and associative : On Q, define a**b=(ab)/2

For the binary operation ** defined below, determine whether ** is commutative and associative : On N, define a**b=a^b

For the binary operation ** defined below, determine whether ** is commutative and associative : On N, define a**b=2^(ab)

For the binary operation ** defined below, determine whether ** is commutative and associative : On Z, define a**b=ab+1

Is the binary operation ** defined on the set of integer Z by the rule a**b=a-b+2 commutative ?