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 given by (a,b) to a + 4b ^(2) is a binary operation.

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.

Show that **: R xxR to R given by a ** b to 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 +: 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.

Show that the relation R in R defined R = {(a, b) : a le b} is reflexive and transitive but not symmetric.

Show that the relation R in the set A={x in Z :0lexle12} is given by R={(a,b) : |a-b|" is a multiple of 4"} is an equivalence relation .

Show that the relation R in the set A={x in z, 0 le x le 12} given by R={(a,b):|a-b| is a multiple of 4} is an equivalence relation.

Consider the binary opertions **: R xx R to R and o : R xx R to R defined as a ** b = |a -b| and a o b =a, AA a, b in R. Show that ** is commutative but not associative, o is associative but not commutative. Further, show that AA a,b,c in R, a ** (b o c) = (a**b) o (a **c). [If it is so, we say that the opertion ** distributes over the opertion 0]. Does o distribute over ** ? Justify your answer.