Show that the `vv: R ->R` given by `(a , b)->m a x {a , b}`and the `^^: R ->R``->`given by `(a , b) ->m in {a , b)`are binary operations.

Show that the VV: RxxR rarr R given by (a, b) rarr max {a, b} and the ^^: RxxR rarr R given by (a, b) rarr min {a, b} are binary operations.

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

Let P be the set of all subsets of a given set X. Show that cup: P xxP rarrP given by (A, B) rarr Acup B and cap : Pxx P rarrP given by (A, B) rarrr A cap B are binary operations on the set P.

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

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.

Show that each of the relation R in the set A = { x in Z : 0 le x le 12} given by R = {(a,b): a = b } is an equivalence relation . Find the set of all elements related to 1 each case.

Show that **: RxxR rarr R defined by a^(**) b = a + 2b is not commutative.

Show that the point P(a, b + c), Q(b, c +a) and R(c, a + b) are collinear.

Determine whether or not each of the definition of ** given below gives a binary operation. In the even that ** is not a binary operation , give justification for this . On R , define ** by a" * " b = ab^2

Determine whether or not each of the definition of ** given below gives a binary operation. In the even that ** is not a binary operation , give justification for this . On Z^(+) , define ** by a" * " b = a - b