Let `M_(2)={:[(x,0),(0,y)]:},x,yinRR-{0}` be the set of `2xx2` matrices, prove that the operation `**` defined on `M_(2)` by `A**B=AB,A,BinM_(2)` is a binary operation.

Let RR be the set of real numbers. Show that the operation ** defined on RR-{0] by a**b=|ab|,a,binRR-{0} is a binary operation on RR-{0}.

Prove that the operation ^^ on RR defined by x^^y= min. of x and y for all x,yinRR is a binary operation on RR .

Show that the binary operation ** defined on RR by a**b=ab+2 is commutative but not associative.

Prove that the operation @ on QQ defined by x@y=(x-2)/(y-2) for all x,yinQQ does not represent a binary operation on QQ .

Let M_(2)(x)d={:{((x,x),(x,x)):},x inRR} be the set of 2xx2 singular matrices. Considering multiplication of matrices as a binary operation, find the identity element in M_(2)(x) . Also find the inverse of an element of M_(2) .

Prove that the binary operation O defined by a o b = a-b+ab, AA a,b in Q is not commutative and associative.

Prove that the binary operation ** on RR defined by a**b=a+b+ab for all a,binRR is commutative and associative.

Prove that the operation @ on QQ , the set of rational numbers, defined by a@b=ab+1 is binary operational on QQ .

Applying set operations, prove that, 2+3 = 5.

The operation ** is defined by a**b=a^(b) on the set Z={0,1,2,3,…} . Show that ** is not a binary operation.