A binary operation `**` is defined on the set `S={0,1,2,3,4}` as follows: `a**b=a+b(mod5)` Prove that `0inS` is the identity element of the binary operation `**` and each element `ainS` is invertible with `5-ainS` being the inverse of the element a.

A binary operation @ is defined on the set A={0,1,2,3,4,5} as follows: a@b=a+b(mod6) for all a,binA. Prove that oinA is the identity element In A is invertible with operation @ and each element A is invertible with 6-ainA being the inverse of the element a.

Prove that 0 is the identity element of the binary operation ** on ZZ^(+) defined by x**y=x+y for all x,yinZZ^(+) .

Define a binary opertion ** on the set {0,1,2,3,4,5} as a**b ={{:(a+b"," , if a + b lt 6), ( a + b -6, if a + b ge 6):} Show that zero is the identity for this opertion and each element ane 0 of the set is invertible with 6-a being the inverse of a.

An operation ** is defined on the set S={1,2,3,5,6} as follows: a**b=ab(mod7) Construct the composition table for operation ** on S and discuss its important properties.

Find the identity element of the binary operation ** on ZZ defined by a**b=a+b+1 for all a,binZZ .

A binary operation ** is defined on the set RR_(0) for all non- zero real numbers as a**b=(ab)/(3) for all a,binRR_(0), find the identity element in RR_(0) .

The binary operation ** on the set A={1,2,3,4,5} is defined by a**b= maximum of a and b. Construct the composition table of the binary operation ** on A.

Prove that the identity element of the binary opeartion ** on RR defined by a**b = min. (a,b) for all a,binRR , does not exist.

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

An operation '*' is defined on a set A={1,2,3,4} as follows: a*b=ab(mod5), all a,b in A ,Prepare the composition table for '*' on A and from the table show that, '*' is a binary opration and '*' is commutative on A.