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.

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 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) .

A binary operation ** is defined on the set ZZ of all integers by a@b=a+b-3 for all a,binZZ . Determine the inverse of 5inZZ .

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.

A binary operaiton ^^ on the set A={1,2,3,4,5} is defined by a^^b=min(a,b) for all a,binA. prepare composition table for the operation ^^ on 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.

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.

An operation ** is defined on the set A={1,2,3,4} as follows: a**b,ab(mod5) for all a,binA .Prepare the compositon table for ** on A and from the table show that - (i) multiplication mod(5) is a binary operation, (ii) ** is commutative on A, (iii) is the identity element for multiplication mod(5) on A, and. (iv) every element of A is invertible.

Let A=NNxxNN , a binary operation ** is defined on A by (a,b)**(c,d)=(ad+bc,bd) for all (a,b),(c,d)inA. Show that ** possesses no identity element in A.

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