Show that `**: R xx R to R` given by (a,b)` to a + 4b ^(2)` is a binary operation.

Show that **: R xxR to R given by a ** b to a + 2b is not associative.

Show that the vv:R xx R given by (a,b) to max {a,b} and the ^^ : R xx R to R given by (a,b) to min {a,b} are binary operations.

Show that +: R xx R to R and × : R xx R to R are commutative binary operations , but - : R xx R to R and div : R _(**) xx R _(**) are not commutative.

Show that **: R × R to R defined by a **b=a + 2b is not commutative.

Let P be the set of all subsets of a given set X. Show that uu:P xx P to P given by (A,B) to A uuB and nn:PxxP to P given by (A,B) to A nnB are binary operations on the set P.

On Z defined * by a ** b = a -b show that * is a binary operation of Z.

The operation * defined a**b = a . Is * a binary operation on z.

Let * be defined on the set on A = {1,2,3,4,5} by a**b = HCF of a and b. Is *a binary operation on A.

Let A be the set of all students of a boys school. Show that the relation R in A given by R = {(a,b):a is sister of b} is the empty relation and R' ={(a,b): the difference between heights of a and b is less than 3 meters } is the universal relation.