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.

Number of binary opertions on the set {a,b} are

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.

Given a non-empty set X, let **: P(X) xx P (X) to P (X) be defined as A **B = (A-B) uu (B-A) , AA A, B in P (X). Show that the empty set phi is the identity for the opertion ** and all the elements A of P (X) are invertible with A ^(-1) =A.

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.

Consider the binary opertion ^^ on the set {1,2,3,4,5} defined by a ^^ b = min {a,b}. Write the opertion table of the opertion ^^.

Given a non-empty set X, consider the binary opertion **: P(X) xx P (Y) to P (X) given by A **B= A nnB AA A, B in P (X), where P(X) is the power set of X. Show that X is the identity element for this opertion and X is the only invertible elemtn in P(X) with respect to the opertion **.

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.

Let ** be the binary opertion on N defined by a **b=H.C.F. of a and b. Is ** commutative ? Is ** associative ? Does there exist identity for this binary opertion on N ?

Let R be the realtion defined in the set A = {1,2,3,4,5,6,7} by R ={(a,b): both a and b are either odd or even}. Show that R is an related to each other and all the elements of the subset {2,4,6} are related to each other, but no element of the subset {1,3,5,7} is related to any element of the subset {2,4,6}.